Cit Universit of New York (CUNY) CUNY Academic Works Open Educational Resources Queensborough Communit College 018 Trigonometr: A Brief Conversation Caroln D. King PhD CUNY Queensborough Communit College Eveln Tam CUNY Queensborough Communit College Fei Ye PhD CUNY Queensborough Communit College Beata Ewa Carvajal CUNY Queensborough Communit College How does access to this work benefit ou? Let us know! Follow this and additional works at: https://academicworks.cun.edu/qb_oers Part of the Mathematics Commons Recommended Citation King, Caroln D. PhD; Tam, Eveln; Ye, Fei PhD; and Carvajal, Beata Ewa, "Trigonometr: A Brief Conversation" (018). CUNY Academic Works. https://academicworks.cun.edu/qb_oers/33 This Tetbook is brought to ou for free and open access b the Queensborough Communit College at CUNY Academic Works. It has been accepted for inclusion in Open Educational Resources b an authorized administrator of CUNY Academic Works. For more information, please contact AcademicWorks@cun.edu.
Trigonometr: A Brief Conversation Caroln King Eveln Tam Fei Ye Beata Ewa Carvajal 018 Edition
These five units are specificall tailored to foster the master of a few selected trigonometr topics that comprise the one credit MA-11 Elementar Trigonometr course. Each unit introduces the topic, provides space for practice, but more importantl, provides opportunities for students to reflect on the work in order to deepen their conceptual understanding. These units have also been assigned to students of other courses such as pre-calculus and calculus as a review of trigonometric basics essential to those courses. This is a first draft of the materials and the are still being edited to improve on the consistenc of the pedagogical approaches and to reflect suggestions from instructors who have used the materials. We are grateful for the support we received from the Open Educational Research (OER) initiative of the Cit Universit of New York (CUNY). We thank ALL of the MA-11 instructors for their invaluable input. Image on cover taken from internet in 017, source unknown. This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike.0 International License. To view a cop of this license, visit http://creativecommons.org/licenses/b-ncsa/.0/.
Contents/ Learning Objectives Unit 1: Angle Measure (page 5) 1.1 Measure in Degrees 1. Measure in Radians 1.3 Conversion from Degrees to Radians 1. Conversion from Radians to Degrees Students will be able to draw angles in standard position, find co-terminal angles, and epress angles in both degree and radian measurement. Unit : Trigonometr of Right Triangles (page 5).1 Sides of a Right Triangle. Definitions of the Trigonometric Functions.3 The Pthagorean Identit. Applications Students will identif the adjacent and opposite sides of an angle, learn the definitions of the si trigonometric functions, recognize the co-functions, construct the two special right triangles, learn fundamental trigonometric identities, and use right triangle trigonometr to solve word problems. Unit 3: Trigonometric Functions of An Angle (page 9) 3.1 Definitions of the Trigonometric Functions of Angles 3. Reference Angles 3.3 Using Reference Angles to Rewrite Trigonometric Functions 3. Finding the Eact Values of Trigonometric Functions Students will learn how to find the si trigonometric functions of an angle, identif reference angles, rewrite trigonometric functions in terms of positive acute angles, and find the eact values of trigonometric functions.
Unit : Graphs of the Sine Curve (page 68).1 The Properties and Graphs of the Sine Curve. Variation I: = A sin.3 Variation II: = sin(b). Variation III: = A sin(b) Students will learn the basic properties of the sine curve, understand the effect of the parameters A and B on the sine curve and graph these curves. Unit 5: Graphs of the Cosine Curve (page 86) 5.1 The Properties and Graphs of the Cosine Curve 5. Variation I: = A cos 5.3 Variation II = cos(b) 5. Variation III: = A cos(b) Students will use their knowledge of co-functions to graph the cosine curve, learn the properties of cosine curves, and understand the effect of the parameters A and B on the cosine curve.
5 Unit 1: Angle Measure The stud of trigonometr begins with the stud of angles. An angle consists of two ras, R 1 and R, with a common endpoint called the verte. We can interpret an angle as a rotation of R 1 onto R. In this case, R 1 is called the initial side and R is called the terminal side. R terminal side rotation direction verte initial side R 1 Figure 1.1 1.1 Measure in Degrees The amount of rotation from the initial side to the terminal side determines the measure of the angle. We use the smbol to show that the angle is measured in degrees. A circle is formed when the terminal side is rotated counterclockwise and ends at the initial side. This circle is considered one complete rotation and the angle is defined to be 360. Some angles are classified b the following special names: θ θ θ θ (a) Acute Angle 0 < θ < 90 (b) Right Angle θ = 90 = 1 rotation (c) Obtuse Angle 90 < θ < 180 (d) Straight Angle θ = 180 = 1 rotation Figure 1. Angles can be positive, negative, and can have multiple rotations. To draw these angles, we use the Cartesian plane, - plane. Each quarter of this plane is called a quadrant and is represented b the Roman numerals, I, II, III, IV. II I O III IV Positions of the four quadrants Figure 1.3 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
6 An angle is in standard position if its verte is at the origin (0, 0) of the - plane and its initial side is fied along the positive -ais. We sa an angle lies in a quadrant if its terminal side is in that quadrant. A positive angle is generated b a counterclockwise rotation of the terminal side. II O 75 I II O 15 I II35 I O II 300 O I III IV III IV III IV III IV (a) (b) (c) Positive angles in standard position Figure 1. (d) Eample 1. Draw each of the angles in standard position. (a) 310 o (b) 165 o (c) 1 o (d) 78 o II I II I II I II I O O O O III IV III IV III IV III IV 1(a) 1(b) 1(c) 1(d) Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
7 A negative angle is generated b a clockwise rotation of the terminal side. II I II I II I II I III O 60 IV III O 150 IV III 15 O IV 30 III O IV (a) (b) (c) Negative angles in standard position Figure 1.5 (d) Eample. Draw each of the angles in standard position. (a) 160 o (b) 50 o (c) 5 o (d) 315 o II I II I II I II I O O O O III IV III IV III IV III IV (a) (b) (c) (d) Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
8 An angle is called quadrantal if its terminal side lies on the -ais or -ais. II I II 90 I II 180 I II 70 I O 0 O O O III IV III IV III IV III IV (a) (b) (c) Quadrantal angles in standard position Figure 1.6 (d) Eample 3. Draw each of the angles in standard position. (a) 360 o (b) 180 o (c) 90 o (d) 70 o II I II I II I II I O O O O III IV III IV III IV III IV 3(a) 3(b) 3(c) 3(d) Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
9 Observations from Eample 3 0 and 360 share the same initial and terminal sides. Which other quadrantal angles drawn in Figure 1.6 and Eample 3 above share the same initial and terminal sides? Two angles with the same initial and terminal sides, but with different rotations, are called coterminal angles. θ = 0 O β = 10 Figure 1.7 Figure 1.7 shows that 0 and 10 are coterminal angles because the have the same initial and terminal sides. Suppose we take the terminal side of the angle which measures 0 and rotate it counterclockwise one full rotation, or 360. θ = 0 O α = 600 Figure 1.8 The terminal side returns to the same position. Therefore, 0 and 600 are coterminal angles. We can generate coterminal angles b adding multiples of 360. 0 + 360 = 600. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
10 Similarl, if we take the terminal side of the angle which measures 0 and rotate clockwise one full rotation, the terminal side returns to the same position (Figure 1.7). This means we can also generate coterminal angles b subtracting multiples of 360. 0 360 = 10. Eample. (a) How man coterminal angles are there for each angle in standard position? (b) Find one other positive and one other negative angle that is coterminal with 0. Eample 5. Find an angle between 0 o and 360 o that is coterminal with the angle 180. We can subtract 360 as man times as we need from 180 to obtain the coterminal angle that we are tring to find. 180 360 = 90 180 ()360 = 180 70 = 560 180 (3)360 = 180 1080 = 00 The angle between 0 o and 360 o that is coterminal with the angle 180 is 00. A more efficient method would be to find how man times 360 divides into 180. Since 360 divides into 180 three times (3 complete rotations) with a remainder of 00, the remainder is the angle we are looking for. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
11 Eample 6. Draw each of the angles in standard position. Find an angle between 0 o and 360 o which is coterminal with each of the given angles. (a) 300 o II I O III IV (b) 50 o II I O III IV (c) 837 o II I O III IV (d) 115 o II I O III IV Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
1 1. Measure in Radians An angle whose verte is at the center of a circle is called a central angle. The radian measure of a central angle θ, is defined as the ratio of the arc length s, and the length of the radius, r. In formula, the radian measure θ of a central angle is θ = s r. θ r s Figure 1.9 When the radius of the circle and the arc length are equal, s = r, we have θ = r r = 1 and we sa the angle θ measures 1 radian. In each of the circles in Figure 1.10, θ is considered to measure 1 radian. θ r = 1 s = 1 s = 3 (a) θ r = 3 (b) Figure 1.10 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
13 We have epressed the ratio of the arc length, s, to radius, r as : θ = s r The arc length of a 360 central angle is the circumference of a circle of radius r. Therefore, the radian measure of a 360 angle is θ = r =. r We now have the following relationship between degrees and radians Equivalentl, 360 = radians. 180 = radians. 1.3 Conversion from Degrees to Radians We can convert from degrees to radians b using or creating a proportion. However, for numbers that are factors of 180, we can simpl use the equivalence radians = 180. Eample 7. (a) Write down ALL the factors of 180. {1,, 3, (b) Complete the table: Degrees Radians 90 3 5 30 5 0 10 15 1 18 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
1 Eample 8. What is 50 in radian measure? We use proportions to do this conversion: 180 = radians If we solve for 50 = radians 50 180 = radians radians = 50 180 radians = 5 18 radians 50 = 5 18 radians The answer 5 radians is considered an eact value. If want an approimate value to the 18 nearest thousandth (rounded to three decimal places) we have 50 0.873 radians. Eample 9. Find the radian measure of 1. 180 = radians If we solve for 1 = radians 1 180 = radians radians = 1 180 radians = 53 5 radians 1 = 53 5 radians 1 3.7 radians. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
15 From our use of proportions, we have arrived at the conversion formula: To convert an angle from degrees to radians we multipl the angle b 180 radians. Eample 10. Convert each angle from degrees to radians. Reduce the fraction and epress our result in terms of (eact value). (a) 5 o (b) 0 o (c) 315 o (d) 1 o Eample 11. Convert each angle from degrees to radians. Round our answer to three decimal places. (a) 153 o (b) 311 o Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
16 1. Conversion from Radians to Degrees We can similarl convert from radians to degrees b proportion. However, if the radian measure contains, we can simpl use the equivalence radians = 180. Eample 1. Convert 5 radians to degrees. 5 radians = 5 (180 ) = 5 Eample 13. Convert radians to degrees. We use proportions to do this conversion: If we solve for 180 = radians = radians 180 = radians radians = (180 ) = 360 This is an eact answer, but what quadrant does the angle lie in? We can find the approimate value b dividing b. = (180 ) = 360 11.591 This angle lies in quadrant II. From our use of proportions, we have arrived at the following conversion formula: To convert an angle from radians to degrees we multipl the angle b 180. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
17 Eample 1. Convert each angle from radians to degrees. (a) 7 (b) 3 (c) (d) 17 6 Eample 15. Convert each angle from radians to degrees. Round to the nearest hundredth. (a) 3.7 radians. (b) 5 radians. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
18 The circles in Figure 1.11 and Figure 1.1 below illustrate the relationship between angles measured in radians and degrees. ( 6 = ) 3 ( 3 6 = ) ( 6 = ) 3 ( ) 5 6 10 90 60 ( 6 ) ( ) 6 6 = 150 180 10 30 360 0 330 ( ) 1 6 = In the Figure 1.11, the circle is divided into increments of 30 or 6 radians. ( ) 7 6 0 70 300 ( ) 11 6 ( 8 6 = ) 3 ( 9 6 = 3 ) Figure 1.11 ( = ) ( 10 6 = 5 ) 3 ( ) 3 ( ) 90 135 5 ( ) = 70 180 360 0 5 315 ( ) 8 = In Figure 1.1, the circle is divided into increments of 5 or radians. ( ) 5 ( ) 7 ( 6 = 3 ) Figure 1.1 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
19 Coterminal Angles in Radians To find angles that are coterminal with an angle in radian measure, we add or subtract multiples of. One convention for epressing the process of creating coterminal angles is the formula, θ + k, where k is an integer. Eample 16. Find angles that are coterminal with θ = 3 When k = 1 we have: When k = we have: θ = 3 + (1) = 3 + = 3 + 6 3 = 7 3 radians radians θ = 3 + () = 3 + = 3 + 1 3 = 13 radians 3 To generate some negative coterminal angles, we let k be a negative integer: When k = 1 we have: θ = 3 + ( 1) = 3 = 3 6 3 = 5 radians 3 II I II I II I II I III O 3 IV III O 7 3 IV III O 13 3 IV III 5 3 O IV (a) (b) Coterminal angles of θ = 3 (c) radians (d) Figure 1.13 Eample 17. Find the angle θ between 0 radians and radians that is coterminal with the angle 11 radians. 3 We can add radians as man times as we need to 11 radians to obtain the coterminal 3 angle that we are tring to find. θ = 11 3 + (1) = 11 3 + 6 3 = 5 3 θ = 11 + () = 11 3 3 + 1 3 = 3 The angle between 0 radians and radians that is coterminal with the angle 11 is 3 radians. 3 radians Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
0 Eample 18. Draw each of the angles in standard position. Find an angle between 0 radians and radians which is coterminal with each of the given angles. (a) 1 5 II I O III IV (b) 7 3 II I O III IV (c) 17 II I O III IV (d) 5 6 II I O III IV Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
1 EXERCISES Practice 1. Convert each angle from degrees to radians. Reduce the fraction and epress our result in terms of. (a) 30 o (b) 160 o (c) 330 o (d) 150 o (e) 70 o (f) 135 o Practice. Convert each angle from degrees to radians. Round our answer to three decimal places. (a) 13 o (b) 37 o Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
Practice 3. Convert each angle from radians to degrees. (a) 7 3 (b) 3 (c) 9 (d) 3 (e) 3 10 Practice. Convert each angle from radians to degrees. Round to two decimal places. (a) 1. radians. (b) radians. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
3 Practice 5. Find a positive angle less than 360 that is coterminal with each angle. (a) 115 (b) 83 (c) 00 (d) 7 Practice 6. Find a positive angle less than radians that is coterminal with each angle. (a) 11 5 (b) 3 6 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
Practice 7. You walk miles around a circular lake. You ma recall that θ = s r. θ r = s = miles What is the measure of the angle θ, in radians that represents our final position relative to our starting position if the radius of the lake is: (Note: Diagrams not drawn to scale.) (a) 1 mile? (b) miles? (c) 3 miles? (d) miles? Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
5 Unit : Trigonometr of Right Triangles.1 Sides of a Right Triangle Trigonometr is the stud of the measurements of a triangle. We start out b concentrating on right triangles and the ratios of the sides of a right triangle A right triangle has one right angle ( 90 or radians) and two acute angles that are complementar to each other. Let s name the two acute angles θ and β. We define the sides (or legs) of the right triangle in relation to angle θ as follows: hpotenuse β opposite leg of θ θ adjacent leg of θ Similarl, the sides of the right triangle in relation to angle β are as follows: hpotenuse β adjacent leg of β θ opposite leg of β As ou can see, the opposite side of θ is the adjacent side of β. And the adjacent side of θ is the opposite side of β. hpotenuse β θ adjacent leg of θ opposite leg of β opposite leg of θ adjacent leg of β Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
6. Definitions of the Trigonometric Functions Now that the sides of a right triangle are defined in relation to an acute angle θ, we can look at trigonometric functions defined as the ratios of the lengths of the sides of the right triangle in relation to angle θ. For eample, sine of θ, denoted b sin θ, is defined as: sin θ = length of the leg opposite θ length of the hpotenuse Eample 19. For the right triangle below, we can determine sin θ: 5 cm hpotenuse θ adjacent leg of θ cm opposite leg of θ 3 cm sin θ = Similarl we can determine sin β: length of the leg opposite θ length of the hpotenuse = 3 cm 5 cm = 3 5. 5 cm hpotenuse β θ opposite leg of β cm adjacent leg of β 3 cm sin β = length of the leg opposite β length of the hpotenuse = cm 5 cm = 5. We notice that the value of a trigonometric function is a number without units, since the function is a ratio of lengths. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
7 There are si trigonometric functions of the angle θ : sine of θ(sin θ), cosine of θ(cos θ), tangent of θ(tan θ), cosecant of θ(csc θ), secant of θ(sec θ), and cotangent of θ(cot θ). The si trigonometric functions are defined in the following chart. sin θ = opposite hpotenuse cos θ = adjacent hpotenuse tan θ = opposite adjacent hpotenuse θ adjacent leg opposite leg csc θ = hpotenuse opposite sec θ = hpotenuse adjacent cot θ = adjacent opposite Eample 0. Based on the right triangle below, find the values of all si functions of both acute angles θ and β. 5 in β 13 in 1 in θ sin θ = 5 13 cos θ = 1 13 csc θ = sin β = csc β = sec θ = cos β = sec β = tan θ = cot θ = tan β = cot β = Observations from Eample 0 (a) sin θ and csc θ are reciprocals. We can see this relationship from the definitions of the functions. Are there other pairs of reciprocals? Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
8 (b) List all the pairs of trigonometric functions of θ that are reciprocals of each other. (c) Compare the values of sin θ and cos β. These values are the same. What is the reason? (d) Are there other functions of θ that are equal to functions of β? Wh do these functions of θ and β have the same value? Do the come in pairs? Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
9 (e) List all the pairs of functions of θ that are equal to the functions of β. Function of θ = Function of β (f) The functions sine and cosine are cofunctions. The name cosine is short for the complement of sine, because θ and β are complementar angles. Eamine the names of the pairs of cofunctions on our list in e above. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
30 Eample 1. Consider the following triangle. Find the values of all si functions of the acute angle θ. A C 6 θ B (a) In order to find the values of all si trigonometric functions of θ, we need the lengths of all the sides of the right triangle. We can find the unknown length AC using the Pthagorean Theorem: a + b = c, where a and b are the lengths of the two legs and c is the length of the hpotenuse. Let s sa the length of AC = b, and appl the Pthagorean Theorem: + b = 6 b = 3 b = 3 b = (b) We now have the lengths of all the sides of the right triangle and we can find the values of the trigonometric functions. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ = Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
31 Eample. Special triangle 5 5 90 (or isosceles right triangles. ) Let s draw three of these 3 1 1 Triangle A 3 Triangle B 3 3 Triangle C These triangles are all of the same shape but the are of different sizes. The are similar triangles and their corresponding sides are proportional. Determine the values of the si trigonometric functions for the acute angle of 5 ( ). Function Values Based on Triangle A (a) First determine the length of the hpotenuse using the Pthagorean theorem. 1 + 1 = c = c = c (b) sin = csc = cos = sec = tan = cot = Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
3 Function Values Based on Triangle B (a) First determine the length of the hpotenuse using the Pthagorean theorem. 3 + 3 = c 18 = c 3 = c (b) sin = csc = cos = sec = tan = cot = Function Values Based on Triangle C (a) First determine the length of the hpotenuse using the Pthagorean theorem. (3 ) + (3 ) = c 36 = c 6 = c (b) sin = csc = cos = sec = tan = cot = Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
33 Observations from Eample - the special triangle 5 5 90 (or ) (a) sin 5 is equal to 1 = based on all three triangles. What is the reason? (b) In general, the values of the trigonometric functions are dependent on the measure of the angle θ and not on the size of the triangle in which angle θ lies. (c) There are man problems that ask us to determine the eact values of functions based on special triangles. We should memorize the angles and their corresponding side of this special triangle 5 5 90 (or ). 1 1 Eample 3. Special triangle 30 60 90 (or 6 3 ) This triangle is half of an equilateral triangle. A 30 30 B 60 1 C 60 1 Using the Pthagorean Theorem, we can determine the third side AC and we have the following special 30 60 90 (or ) triangle. We should memorize the angles and their 6 3 corresponding side of this special triangle. A 30 3 B 60 1 C Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
3 Determine the si functions for 30 ( 6 ) and 60 ( 3 ). sin 30 = 1 csc 30 = sin 60 = csc 60 = cos 30 = 3 sec 30 = cos 60 = sec 60 = tan 30 = cot 30 = tan 60 = cot 60 = Observations from Eample 3 - the special triangle 30 60 90 (or 6 3 ) (a) Compare tan to cot. Wh are the the same? 3 6 (b) List all the pairs of cofunctions. (c) List all the pairs of functions that are reciprocals of each other. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
35 (d) Check that sin 3 cos 3 = tan 3. (e) Does the relationship in (d) above hold for the angle 6? (f) Does the relationship in (d) above hold for the angle? (g) Does the relationship in (d) above hold for an angle θ? Eamine the definitions of sin θ, cos θ, and tan θ. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
36.3 The Pthagorean Identit The Pthagorean Theorem a + b = c is in terms of the lengths of the sides of a right triangle. This theorem ma also be epressed in terms of trigonometric functions of an acute angle θ of a right triangle. sin θ + cos θ = 1 This is an identit because this equation is true for all values of θ, for which each of the trigonometric functions is defined. We can easil derive this Pthagorean Identit. Let s consider this right triangle: B 1 a A b C where the length of the two legs are a and b and the hpotenuse is 1. The Pthagorean Theorem as applied to this triangle is a + b = 1. From the triangle, we have sin θ = cos θ = Let s rewrite the Pthagorean Theorem in terms of sin θ and cos θ. a + b = 1 sin θ + cos θ = 1 Variations of The Pthagorean Identit Identit: There are two variations of the Pthagorean tan θ + 1 = sec θ 1 + cot θ = csc θ Let s derive tan θ + 1 = sec θ. The other variation is left as an eercise. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
37 To derive this variation, we start out with the Pthagorean Identit we just learned. sin θ + cos θ = 1 We ll divide both sides of the equation b cos θ. The result is: sin θ cos θ + cos θ cos θ = 1 cos θ We have just derived the identit : tan θ + 1 = sec θ. The chart below summarizes the trigonometric identities that we ve learned so far. Reciprocal Identities Fundamental Trigonometric Identities Quotient Identities sin θ = 1 csc θ csc θ = 1 sin θ cos θ = 1 sec θ sec θ = 1 cos θ tan θ = 1 cot θ cot θ = 1 tan θ Pthagorean Identities tan θ = sin θ cos θ cot θ = cos θ sin θ sin θ + cos θ = 1 tan θ + 1 = sec θ 1 + cot θ = csc θ Cofunction Identities In radians ( ) ( ) sin θ = cos θ cos θ = sin θ ( ) ( ) tan θ = cot θ cot θ = tan θ ( ) ( ) sec θ = csc θ csc θ = sec θ In degrees sin (90 θ) = cos θ cos (90 θ) = sin θ tan (90 θ) = cot θ cot (90 θ) = tan θ sec (90 θ) = csc θ csc (90 θ) = sec θ Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
38. Applications Right triangle trigonometr is often used to measure distance and height. Eample. From a point on the ground 500 feet from the base of a building, a surveor measures the angle of elevation from the point on the ground to the top of the building to be 5. Determine the height of the building, to the nearest foot. Solution. (A) We first draw a picture based on the information given b the problem. 5 line of sight 500 feet feet The angle of elevation is the angle formed b the horizontal ground and the line of sight from the point on the ground to the top of the building. (B) We now have a right triangle. We are looking for the height of the building and so we let it be feet. (C) In the triangle, we want to relate the given angle of 5 to the sides that measure 500 feet and feet. For the 5 angle, the side measuring 500 feet is the adjacent side and the side measuring feet is the opposite side. Now, what is the trigonometric function that is defined b the opposite and adjacent sides? (D) tan 5 = 500 = 500 tan 5 use our calculator to find tan 5 33 feet rounded to the nearest foot Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
39 Eample 5. Determine the length of AC to the nearest tenth of a meter. A meters 0 C 00 meters B Solution. (A) Relate the sides measuring 00 meters and meters to the 0 angle. The side BC measuring 00 meters is the The side AC measuring meters is the of the 0 angle. of the 0 angle. What is the appropriate trigonometric function? (B) cos 0 = 00 = 00 cos 0 153. meters Eample 6. A road is inclined at an angle of 5. A bicclist travels 5000 meters along the road. How much has her altitude increased? Round to the nearest meter. Solution. (A) Draw a picture 5000 meters 5 meters (B) Relate 5000 meters and meters to the angle of 5. What is the appropriate function? = sin 5 5000 (C) Solve the equation = 5000 sin 5 36 meters. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
0 Eample 7. Determine the angle θ to the nearest degree. A 50 feet B 90 feet θ C Solution. (A) Relate the side AB to angle θ and the side AC to right angle B. Side AB is 50 feet long and is the side of angle θ. Side AC is 90 feet long is the of the triangle. What is the appropriate trigonometric function? (B) sin θ = 50 90 sin θ = 0.556 We are looking for the acute angle, θ whose sine value is 0.556. The mathematical notation is: θ = sin 1 0.556, or 1 50 θ = sin 90 On the calculator, the second function (nd) of sin is sin 1 which gives the acute angle when rounded to the nearest degree. θ = 3 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
1 EXERCISES Practice 8. List all the pairs of functions that are reciprocals of each other. Practice 9. Find the cofunction with the same value as the given function. Function = Cofunction a) sin 5 = b) cot 78 = c) cos 17.6 = d) sec 8 = e) csc 5 = f) tan 3 7 = Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
Practice 10. Use a calculator to find the values of trigonometric functions and round our answer to decimal places. (a) sin 37 (b) cos 59 (c) tan 1.8 (d) sec 35 (e) csc 1. radians (f) cot 0.8 radians Practice 11. Use a calculator to find the acute angles to the nearest tenth of a degree. (a) sin 1 (0.316) (b) tan 1 (1.351) (c) cos 1 (0.7618) Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
3 Practice 1. Find the unknown sides and, to the nearest thousandth. Please note that the triangles are not drawn to scale. 3 o (a) 8 (b) 5 o 5 (c) 7 38 o Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
Practice 13. Angle θ is an acute angle and cos θ = 1. Find the values of the other five 5 trigonometric functions. Practice 1. A house is on the top of a hill with a road that makes an angle of inclination of 10. The road is 1000 feet long from the bottom of the hill to the house. What is altitude of the house, to the nearest foot? Practice 15. At a certain time of da, an 85 foot tall tree casts a 35 foot long shadow. Find the angle of elevation of the sun to the nearest tenth of a degree. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
5 Practice 16. For acute angle A, sin A = 0.7 and cos A = 0.51. (a) Use a quotient identit to find tan A. (b) Use a reciprocal identit to find sec A. Practice 17. From the top of a 65 foot-high building, an observer measures the angle of depression of a biccle on the sidewalk as 5 o. Find the distance from the biccle to the base of the building to the nearest foot. Practice 18. John wants to measure the height of the tree he planted man ears ago. He walks eactl 18 meters from the base of the tree, turns around and looks up to the top of the tree. The angle of elevation from the ground to the top of the tree is 7 o. How tall is the tree to the nearest tenth of a meter? Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
6 Practice 19. Angle θ is an acute angle and sin θ = 6 7. (a) Find cos θ using a Pthagorean identit. (b) Find the values of all the other trigonometric functions using quotient identities and reciprocal identities. (c) Are there others was to solve this problem? Compare this eercise with Practice 13. (d) Determine the measure of θ to the nearest degree. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
7 Practice 0. A 5-foot ladder leans against a wall. The bottom of the ladder is 18 feet from the base of the wall. (a) Determine the angle the bottom of the ladder makes with the ground. Round to the nearest degree. (b) Determine the angle the top of the ladder makes with the wall. Round to the nearest degree. Determine this angle using two different methods. (c) How high, to the nearest tenth of a foot, is the top of the ladder from the ground? Is there more than one method to determine the height? Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
8 Practice 1. A group of students wants to measure the width of a river. The set up a post P on their side of the river directl across from a tree T. Then the walked downstream 50 feet and measured the angle formed b the line of sight to the post P and the line of sight to tree T. The found the angle is 66. Determine the width of the river. Round our answer to the nearest foot. T P 66 50 feet Practice. The straight line = forms an angle with the positive -ais in the first quadrant. Let s call this angle θ. Determine the value of tan θ. Compare tan θ to the slope of the line =. Wh are the the same? Practice 3. Based on the Pthagorean Identit: sin θ + cos θ = 1 derive the Pthagorean identit 1 + cot θ = csc θ Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
Unit 3: Trigonometric Functions of An Angle To etend the definition of the si trigonometric functions, we return to the Cartesian Plane and angles in standard position. 3.1 Definition of The Trigonometric Functions of Angles Trigonometric Functions of Angles Let θ be an angle in standard position and P = (, ) be a point on the terminal side of θ. Using the Pthagorean Theorem, the distance between P and the origin is given b r = +. The si trigonometric functions are defined as follows. 9 sin θ = P (, ) r cos θ = r θ r tan θ =, ( 0) csc θ = r, ( 0) sec θ = r, ( 0) cot θ =, ( 0) We can use the definitions of the sine and cosine functions to verif the Pthagorean Identit: Since sin θ = r and cos θ = r ( r sin θ + cos θ = 1 ) + ( r ) = ( + r ) = r r = 1 The value of each of the si trigonometric functions depends on the values of, and r. The distance, r, between the origin and Point P is alwas a positive value. The quadrant that θ lies in determines whether the value of the si trigonometric functions is positive or negative. Quadrant I In this quadrant, > 0 and > 0. Since, and r are all positive, all si trigonometric functions are positive. r θ P (, ) Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
50 Quadrant II In this quadrant, < 0 and > 0. P (, ) r θ sin θ = r > 0 csc θ = r > 0 cos θ = r < 0 sec θ = r < 0 tan θ = < 0 cot θ = < 0 So in quadrant II, sin θ and its reciprocal csc θ are both positive. The other four trigonometric functions are negative. Quadrant III In this quadrant, < 0 and < 0. θ r P (, ) sin θ = r < 0 cos θ = r < 0 csc θ = r sec θ = r cot θ = tan θ = Which trigonometric functions are positive? Which trigonometric functions are negative? Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
51 Quadrant IV In this quadrant, > 0 and < 0. θ r P (, ) sin θ = r < 0 cos θ = r > 0 csc θ = r sec θ = r cot θ = tan θ = Which trigonometric functions are positive? Which trigonometric functions are negative? We can use the mnemonic All Students Take Calculus (ASTC) to help us recall the signs of the values of the si trigonometric functions in each of the quadrants. The "A" is in quadrant I and means that all si trigonometric functions are positive. The "S" is in quadrant II and means that sin and its reciprocal are positive. The "T" is in quadrant III and means that tan and its reciprocal are positive. The "C" is in quadrant IV and means that cos and its reciprocal are positive. S A T C ASTC Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
5 Eample 8. What quadrant does θ lie in if cos θ > 0 and sin θ < 0? Step 1. Find the quadrants in which cos θ > 0. cos is positive in Quadrants I and IV. Step. Find the quadrants in which sin θ < 0. sin is negative in Quadrants III and IV. Step 3. Find the quadrant that satisfies both conditions θ lies in Quadrant IV Eample 9. Find the quadrant in which θ lies. (a) cos θ > 0 and tan θ > 0 (b) csc θ < 0 and sec θ < 0 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
53 Eample 30. Let P = ( 3, ) be a point on the terminal side of an angle θ. Find the eact value of the si trigonometric functions of θ. Step 1. Find the radius r using the Pthagorean Theorem: r = +. = P ( 3, ) r = 5 θ = 3 r = + = ( 3) + = 5. Step. Evaluate the functions using definitions. sin θ = r = 5 cos θ = r = 3 5 tan θ = = 3 = 3 csc θ = 1 sin θ = 5 sec θ = 1 cos θ = 5 3 cot θ = 1 tan θ = 3 and sin θ < 0, find the eact values of the ALL si trigono- Eample 31. Given tan θ = 1 5 metric functions of θ. Step 1. Find the quadrant in which θ lies. Since tan θ > 0 and sin θ < 0, θ is in Quadrant III. In this quadrant, < 0 and < 0 so we epress tangent as the ratio of two negative numbers and write: tan θ = 1 5. Step. Appl the Pthagorean theorem with appropriate signs to find r. r = + = ( 5) + ( 1) = 13. = 5 θ = 1 r = 13 P ( 5, 1) Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
5 Step 3. Evaluate the functions using definitions. sin θ = r = 1 13 = 1 13 cos θ = r = 5 13 = 5 13 tan θ = = 1 5 csc θ = 1 sin θ = 13 1 sec θ = 1 cos θ = 13 5 cot θ = 1 tan θ = 5 1 Eample 3. Find the eact value of the All si trigonometric functions. (a) tan θ = 3 and cos θ < 0. (b) Given the point (3, 3) on the terminal side of an angle. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
55 3. Reference Angles We use reference angles to simplif the calculation of trigonometric functions of non-acute angles. The trigonometric function for a non-acute angle has the same value, ecept possibl for the sign, as the corresponding function of the acute angle. The acute angle is called the reference angle of the non-acute angle. Let θ be an angle in standard position. The reference angle θ associated with θ is the acute angle formed b the terminal side of θ and the -ais. II III I θ = θ IV II θ III θ I IV II III θ θ I IV II III θ θ I IV Finding Reference Angles Positive angles with corresponding reference angles Method 1 Draw the angle in standard position and find the acute angle formed b the terminal side of θ and the -ais. Method (Without drawing the angle.) Step 1. Find the quadrant in which θ lies. Step. Use the table below to calculate the reference angle, θ. Recall that 360 = radians and that 180 = radians. Step 3. If θ is negative or larger than radians (or 360 ) find the coterminal angle that is between 0 and (or between 0 and 360 ). Recall that coterminal angles have the same initial and terminal sides, but possibl different rotations. Quadrant Containing θ Reference angle θ Reference angle θ (degrees) (radians) I θ = θ θ = θ II θ = 180 θ θ = θ III θ = θ 180 θ = θ IV θ = 360 θ θ = θ Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
56 Eample 33. Find the reference angle θ, for each given angle, θ. Draw each angle in standard position and find the acute angle formed b the terminal side of θ and the -ais. 1. If θ = 160, then θ = II θ θ = 160 I III IV. If θ = 31, then θ = II I III IV 3. If θ = 7, then θ = II I III IV. If θ = 39, then θ = II I III IV Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
57 Eample 3. Find the reference angle θ for the angle θ. (a) θ =. 5 Since θ is in the first quadrant, the reference angle is θ = θ =. 5 (b) θ = 150. Since θ is in the second quadrant, the reference angle is θ = 180 150 = 30. Converting θ = 150 to radians, we obtain θ = 5 6 = 6. (c) θ = 5. Since θ is in the third quadrant, the reference angle is θ = 5 180 = 5. Converting θ = 5 to radians, we obtain θ = 5 =. (d) θ = 300. Since θ is in the fourth quadrant, the reference angle is θ = 360 300 = 60. Converting θ = 300 to radians, we obtain θ = 5 3 = 3. (e) θ = 7 6 radians. Since θ is in the third quadrant, the reference angle is θ = 7 6 = 6. (f) θ = 3 radians. Since θ is in the second quadrant, the reference angle is θ = 3 =. Eample 35. Let θ be an angle in standard position. Find the reference angle θ for each. (a) θ = 18 (b) θ = 75 (c) θ = 131 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
58 (d) θ = 50 (e) θ = 3 (f) θ = 7 (g) θ = 17 3.3 Using Reference Angles to rewrite trigonometric functions as functions of positive acute angles. Eample 36. Rewrite tan 15 as a function of a positive acute angle. DO NOT EVALUATE. Step 1. Step. Step 3. Find the quadrant in which 15 lies. 15 in Quadrant II Find the sign of tan in that Quadrant. tan is negative in Quadrant II. Find the reference angle in Quadrant II, θ = 180 θ = 180 15 = 55 Step. Finall, we can we write: tan 15 = tan 55 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
59 Eample 37. Rewrite the epression as a function of a positive acute angle. DO NOT EVALUATE. (a) sin 311 = (b) cos 1 = (c) csc 15 = (d) tan 609 = (e) sin ( ) 7 = 1 ( ) 13 (f) cot = 9 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
60 3. Using reference angles and special right triangles to FIND THE EXACT VALUE of Trigonometric Functions. A A 30 3 1 5 B 60 1 C C 1 5 B You ma recall: 30 = 6 radians 5 = radians 60 = 3 radians Eample 38. Find the EXACT VALUE of sin ( ). 3 Step 1. Step. Step 3. Find the quadrant in which 3 lies. in Quadrant III 3 Find the sign of sin in that Quadrant III. sin is negative in Quadrant III. Find the reference angle in Quadrant III. θ = 3 = 3 Step. We use the special triangle ( to ) evaluate the function. ( ) 3 sin = sin = 3 3. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
61 Eample 39. Find the EXACT VALUE of each trigonometric function. (a) tan 10 = (b) cot 315 = (c) csc 150 = (d) cot 0 = (e) cot ( ) 5 = (f) sin ( ) 7 = Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
6 EXERCISES Practice. Find the quadrant in which θ lies. (a) tan θ < 0 and sec θ > 0. (b) cos θ < 0 and csc θ > 0. Practice 5. Find the eact value of the All si trigonometric functions. (a) cos θ = 1 3 and θ is in quadrant IV. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
63 (b) sin θ = 8 17 and cos θ > 0. (c) Given the point ( 1, ) on the terminal side of an angle. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
6 Practice 6. Let θ be an angle in standard position. Find the reference angle θ for each. (a) θ = 17 (b) θ = 13 (c) θ = 3 (d) θ = 665 (e) θ = 7 6 (f) θ = 11 6 (g) θ = 6 5 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
65 Practice 7. Rewrite the epression as a function of a positive acute angle. DO NOT EVALUATE. (a) tan 80 = (b) cot 195 = (c) sin 39 = (d) tan ( ) 6 = 5 (e) cos ( ) = 5 ( ) 11 (f) csc = 9 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
66 Practice 8. Find the EXACT VALUE of each trigonometric function. (a) sin 10 = (b) cos 330 = (c) sec 300 = (d) sin ( ) 3 = (e) cos ( ) = 3 (f) tan ( ) = 3 (g) sin ( ) 5 = 3 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
67 Practice 9. Find the equivalent epressions without using our calculator! (a) Which epression(s) is(are) equivalent to sin 00? (i) sin 0 (ii) cos 0 (iii) cos 70 (iv) sin 70 (b) Which epression(s) is(are) equivalent to cos 150? (i) cos 30 (ii) sin 60 (iii) cos( 10 ) (iv) cos 10 (c) Which epression(s) is(are) equivalent to tan 30? (i) cot 0 (ii) cot 0 (iii) tan 50 (iv) tan 50 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
68 Unit : The Basic Sine Curve: = sin.1 The Properties and Graphs of the Sine Curve (a) Let be an angle in terms of radians. (b) Because the values of sin repeat ever radians, the sine function is a periodic function. (c) The period of the function is the length of the interval of that is needed to produce one complete ccle of the curve. (d) The sine curve oscillates about a horizontal line (let s call it the center-line). For the basic sine curve, it is the -ais. (e) The amplitude of the function is the distance from the center-line of the function to the maimum value. (Or from the center-line of the function to the minimum value.) (Or one-half the difference between the maimum value and the minimum value.) Since the amplitude is a distance, it is a positive number. (f) The tables of values of the basic sine curve = sin. in Quadrant I Degrees Radians Reference Angle = sin 0 0 0 0 30 6 30 1 5 5.71 60 3 60 3.87 90 90 1 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
69 in Quadrant II Degrees Radians Reference Angle = sin 10 3 60 sin 60 = 3.87 135 3 5 sin 5 =.71 150 5 6 30 sin 30 = 1 180 0 sin 0 = 0 in Quadrant III Degrees Radians Reference Angle = sin 10 7 6 30 sin 30 = 1 5 5 5 sin 5 =.71 0 3 60 sin 60 = 3.87 70 3 90 sin 90 = 1 in Quadrant IV Degrees Radians Reference Angle = sin 300 5 3 60 sin 60 = 3.87 315 7 5 sin 5 =.71 330 11 6 30 sin 30 = 1 360 0 sin 0 = 0 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
70 Quadrant I The graph of = sin Quadrant II Quadrant III Quadrant IV 1 3 1 30 6 5 60 3 90 10 135 150 3 3 5 6 180 10 5 0 7 6 5 3 70 3 300 315 330 5 3 7 11 6 360 1 3 1 1 ccle period: Observations from the graph of = sin : (a) The curve of Quadrant I is smmetrical to Quadrant II about the vertical line =. (b) The curve of Quadrant III is smmetrical to Quadrant IV about the vertical line = 3. (c) What is the horizontal line about which the sine curve oscillates (the center-line)? (d) Compare the curve of Quadrants I and II to the curve of Quadrants III and IV relative to the center-line. (e) The graph of the sine curve highlights the fact that the values of sine are positive in Quadrants I and II, and the values are negative in Quadrants III and IV. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
71 Questions on the graph: (a) What is the period of the graph? (b) What is the amplitude of the graph? (c) What are the coordinates of the maimum and minimum points? (d) What is the center-line of the graph? What are the coordinates of the points of the graph that lie on the center-line? (e) Wh are the values of the sine curve positive in Quadrants I and II? (f) Wh are the values of the sine curve negative in Quadrants III and IV? (g) In this course, we are asked to roughl sketch a complete ccle of a sine curve. To do this, we just consider the following 5 points of a sine curve: the maimum and the minimum points above and below the center-line and the three points that are on the center-line of the curve. These points are given in the table below. List the coordinates of these 5 points, plot them on the graph and also label these points on the graph. Do ou see how these points give ou a rough sketch of the sine curve? (h) What is the relationship between the -coordinates of the five points and the period of the curve? = sin (, ) 0 = sin 0 = 0 = sin = 1 (ma) = sin = 0 3 = sin 3 = 1 (min) = sin = 0 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
7. Variation I: = A sin Consider the function = A sin, where A is a constant. In this case, all the values of the basic sine curve are multiplied b the constant A. Eample 0. Compare the basic graph = sin to the graphs = sin and = 3 sin. To roughl sketch these graphs, we are plotting the 5 points. The -value of ever point is multiplied b the constant A. = sin = sin = 3 sin 0 0 0 0 = 0 0? 1 1 =? 0 0 = 0? 3 1 3 ( 1) = 3? 0 0 = 0? Write the correct equation net to each graph. 3 1 3 5 3 7 1 3 Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
73 Observations from the graph of = A sin : How does multipling sin b A change the following characteristics of sin : Period: Amplitude: Center-Line: Eample 1. For each of the three curves: = sin, = sin, and = 3 sin, state the period, amplitude and the center-line. = sin = sin = 3 sin Period: Period: Period: Amplitude: Amplitude: Amplitude: Center-Line: Center-Line: Center-Line: Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
7 Eample. (a) Roughl sketch one complete ccle of = 1 sin. On the graph, label the coordinates of the 5 points. Determine the period, the amplitude and the center-line of the curve. (b) The -coordinates of the 5 points are related to the period of the graph. State the - coordinates in terms of the period. (c) The -coordinates of the 5 points are related to another feature of the graph. State the -coordinates in terms of that feature. = 1 sin (, ) Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
75 Eample 3. Roughl sketch one complete ccle of = sin. On the graph, label the coordinates of the 5 points. Determine the period, the amplitude and the center-line of the curve. How does a negative constant A change the sine curve? = sin (, ) Eample. Briefl describe what A in = A sin does to the basic sine curve. How is the graph of the basic sine curve changed when it is multiplied b A? What features are not changed b the multiplication b A? Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
76 Eample 5. Match each equation with its graph. Also write the period, amplitude and center-line net to each graph. (a) = sin (b) = sin (c) = sin 3 5 3 7 (d) = 1 sin (e) = sin A 1 1 1 1 3 5 3 7 3 5 3 7 1 1 1 1 B C 3 3 1 1 3 5 3 7 3 5 3 7 1 1 3 D 3 E Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
77.3 Variation II: = sin B Consider the function = sin B, where B is a constant. Eample 6. Graph the function = sin. Let s look at some values of the graph. in radians = sin 0 sin 0 = 0 sin( ) = sin = 1 1 1 6 sin( ) = sin = 3 6 6 3 sin( ) = sin = 1 sin( ) = sin = 0 sin = 0 We observe that when goes from 0 to, the corresponding value of sin goes from sin 0 to sin, forming one complete ccle of the sine curve. This observation can be stated in other was: (a) Because we are considering the sine of times the angle, the sine values go twice as fast. A complete ccle from sin 0 to sin is achieved in half of a regular period of. (b) The period of the curve = sin is half of the regular period of or =. (c) Because the sine values go twice as fast, there are complete ccles in a regular period of. Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
78 Write the correct equation net to each graph. 1 3 5 3 7 1 Observations from the graph of = sin : (a) The graph of = sin completes full ccles in. The graph of = sin completes one full ccle in. (b) How does multipling the angle b change the following characteristics of sin : Period: Amplitude: Center-Line: Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
79 Eample 7. Graph the function = sin. The value is the sine value of times the angle. Therefore one complete ccle of sine values is reached times as fast. Let s use the following questions to help us graph the function. (a) What is the period of = sin? (b) What is the amplitude? (c) What is the center-line? (d) How man complete ccles of = sin are there in? (e) What are the -coordinates of the 5 points? (f) Roughl sketch one complete ccle of = sin. Notice how the -coordinates of the 5 points are related to the period and how the -coordinates are related to the amplitude. = sin (, ) 0 1 ( ) = 8 ( ) = 3 ( ) = 3 8 ( ) = Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal
80 Eample 8. Briefl describe how B in = sin B affects the Period: Amplitude: Center-Line: We ve observed that the period of the curve = sin is half of the regular period of or =. Similarl, we observed that the period of the curve = sin is one-quarter of the regular period of or =. We can etend this concept to the curve = sin B, and arrive at the following formula to find the period, P : P = B The constant B is called the angular frequenc and represents the number of ccles of the curve that is completed between 0 and. Eample 9. (a) Roughl sketch one complete ccle of = sin 1. On the graph, label the coordinates of the 5 points. Determine the period, amplitude and the center-line of the curve. (b) Write the -coordinates of the 5 points in terms of the period. (c) Write the -coordinates in terms of the amplitude. = sin 1 (, ) Trigonometr: A Brief Conversation b King, Tam, Ye, and Carvajal