Algebra2/Trig Chapter 10 Packet In this unit, students will be able to: Convert angle measures from degrees to radians and radians to degrees. Find the measure of an angle given the lengths of the intercepted arc and radius Find trigonometric function values in radian measure Use Pythagorean Identities to simplify trigonometric expressions Use Pythagorean Identities to find function values Identify the domain and range of trig functions. Evaluate inverse trig functions Name: Teacher: Pd: 1
Algebra 2 Trig Chapter 10 Homework Sheet Assignment # Pages Text reference Day 1 Pages 404 405 Pages 400-406 10-1 Radian Measure #3-41 odd Day 2 10-2 Trigonometric Function Values and Radian Measure Pages 409 410 3 14 all, 25 28 all, 32 Pages 406-409 Day 3 10-3 Pythagorean Identities Pages 414 3 13 odd, 15-22 Pages 411-414 Day 4 10-4 Domain and Range of Trigonometric Functions 10-5 Inverse Trig Functions See Attached On Pages #29 31 in this packet Pages 414-423 Day 5 10-6 Cofunctions Page 427 # s 3-23 odd Pages 425-427 2
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Day 1 - Ch. 10-1: Radian Measure SWBAT: (1) convert angle measures from degrees to radians and radians to degrees. (2) find the measure of an angle given the lengths of the intercepted arc and radius Radians are another way of measuring angles. A radian is the unit of measure of a central angle that intercepts an arc equal in length to the radius of the circle. 360 o = 2πr 360 o = 2π(1) r = 1 in a unit circle 360 o = 2π 180 o = π In conclusion, π radians = 180 o. Furthermore, if 360 o = 2π radians, then: and (divided by 360 and simplified) (divided by 2π and simplified) 6
Concept 1: Converting between Radians and Degrees Changing Radians to Degrees To convert radians to degrees, we make use of the fact that π radians equal s one half circle, or 180º. This means that if we divide radians by π, the answer is the number of half circles. Multiplying this by 180º will tell us the answer in degrees. So, to convert radians to degrees: or 1) Find the degree measure of an angle of 4 radians. 2) Find the degree measure of an angle of 3 radians. Changing Degrees to Radians To convert degrees to radians, first find the number of half circles in the answer by dividing by 180º. But each half circle equals π radians, so multiply the number of half circles by π. So, to convert degrees to radians: or 1) Express in radian measure an angle of 75. 2) Express in radian measure an angle of 120 o. 7
You Try it! Find the radian measure of an angle given the degree measure: 1) 30 2) 45 3) 120 4) 220 5) 100 6) 300 *******Convert to radians and MEMORIZE the following******** 7) 0 8) 90 9) 180 10) 270 11) 360 Find the degree measure of each angle given the radian measure: 12) 3 13) 5 6 14) 9 15) 7 2 16) 2 5 17) 2 3 *****Convert and MEMORIZE the following******* 18) 0 19) 2 20) π 21) 3 2 22) 2π 8
Concept 2: Measure of an Angle in Radians To find the measure of an angle in radians when you are given the lengths of the arc and radius: Measure of an angle in radians = length of the intercepted arc length of radius B In general, if Ө is the measure of a central angle in radians, s is the length of the intercepted arc, and r is the length of a radius, then: O A Ө = r s If both members of this equation are multiplied by r, the rule is stated s = Ө r Examples 1) In a circle, the length of a radius is 4cm. Find the length of an arc intercepted by a central angle whose measure is 1.5 radians. 2) A 20" pendulum swings through an angle of 1.5 radians. What is the distance covered by the tip of the pendulum? 3) Find the length of the arc when Ө = 55 and the radius is 1.25. 9
You try it! 4) A circle has radius 1.7 inches. Find the length of an arc intercepted by a central angle of 2 radians. 5) A ball rolls in a circular path that has a radius of 5 inches, as shown in the accompanying diagram. If the ball rolls through an angle of 2 radians, find the distance traveled by the ball. 2 radians 5 6) A ball is rolling in a circular path that has a radius of 10 inches, as shown in the accompanying diagram. What distance has the ball rolled when the subtended arc is 54? Express your answer to the nearest hundredth of an inch. 10
Summary/Closure Converting Radians to and from Degrees Since 180º degrees is equal to radians, then a Fancy Form of 1 that can be used to change degrees to radians is: Example 1: Convert 210º to radians Example 2: Convert radians to degree Exit Ticket 11
Day 2: 10-2 Trigonometric Function Values and Radian Measure SWBAT: find trigonometric function values in radian measure Do Now: Common Angles in Radian Form: MEMORIZE THESE π rad = rad = rad = rad = rad 2 3 4 6 Degrees 0 30 45 60 90 180 270 Radian Sin Cos Tan 12
Trigonometric Functions involving Radian Measure Since angle measure can be expressed in radians as well as in degrees, we can find values of trigonometric functions of angles expressed in radian measure. To do this, we convert the radian measure to a degree measure and follow the procedures learned earlier. 4 Example: Find the exact value of sin. 3 Step 1: Change to degrees: Step 2: Draw a unit circle with an angle of o. Step 3: Find the reference angle: Step 4 : Find the exact value of the function of the reference angle. 13
Examples - Find the exact value of each of the following: 2 1) cos 3 3 2) tan 4 5 3) sin 2 4) sin 3 4 5) cos 3 6) sin 6 7) tan 5 6 8) sin 3 2 9) cos ( ) 14
15 10. If a function f is defined as f(x) = cos 2x + sin x, find the numerical value of f( 2 ). 11. Find the numerical value of f(x) = 3cosx sin2x of f( ) for the given function f. 12. Find the numerical value of f(x) = 2sinx + 2cosx of f function f. 3 for the given 15
16 Summary/Closure Exit Ticket 16
17 Day 3 - Chapter 10 Section 3: Pythagorean Identities SWBAT: use Pythagorean Identities to (1) simplify trigonometric expressions (2) find function values Warm - Up Given the unit circle with equation x 2 + y 2 =1, we know x = and y =. Therefore, (cos ) 2 + (sin ) 2 = 1 We can write (cos ) 2 as cos 2 and (sin ) 2 as sin 2. We can rewrite the above equation as cos 2 + sin 2 = 1. This equation is called an identity. An identity is an equation that is true for all values of the variable for which the terms of the variable are defined. Specifically, the above identity is called a Pythagorean Identity since it is based on the Pythagorean Theorem. 17
18 Example: Verify that cos 2 + sin 2 = 1 Now take the Pythagorean Identity cos 2 + sin 2 = 1 Divide it through by cos 2 Divide it through by sin 2 18
19 Rules of multiplication, division, addition and subtraction can be applied: Example 2: Simplify by factoring cos 2 + cos = Example 3: Simplify by factoring 1 sin 2 = Example 4: Simplify Example 5: Express sec cot as a single function. 19
20 Example 6: Write the expression 1 + cot 2 in terms of sin, cos, or both. Example 7: Show that (1 cos )(1 + cos ) = sin 2. Example 8: a) If cos = and is in the fourth quadrant, use an identity to find sin. b) Now find: 1) tan 2) sec 3) csc 4) cot 20
21 Example 9: If tan A = and sin A < 0, find cos A. Summary/Closure Exit Ticket 21
22 Day 4: 10-4&5 Domain and Range/Inverse Trig Functions SWBAT: (1) Evaluate inverse trig functions (2) Identify the domain and range of trig functions. Do Now: Find in degrees, if 0 o < 360 o : sin = ½ 22
23 Concept 1: Domain and Range of the trig functions. Sine and Cosine What numbers are we allowed to put into f(x) = sin(x) What numbers are we allowed to put into f(x) = cos(x) Therefore, the domain of sine and cosine is. The range of both sine and cosine are different. As we get our y-coordinates from the unit circle, as we rotate around, what is the largest and smallest values we get for the y s? Therefore, the range of sine and cosine is. Tangent and Secant Tangent is different because of its nature. tan(x) = sec(x) = Therefore, the domain of tangent and Secant is. Therefore, the range of tangent is. 23
To find its range were going to use a little inequality we know from cos(x). We know the range of the cos(x) is between [-1,1] we are going to split it and do some algebra on it to figure out what sec(x) must be. 0 cos( x ) 1 or 1 cos( x ) 0 24 Therefore, the range of secant is. Cotangent and Cosecant cot(x) = csc(x) = Therefore, the domain of tangent and Secant is. Cotangent is similar to tangent, Therefore, the range of cotangent is Cosecant is similar to secant, It s range can be found exactly the same as we did in sec(x), Therefore, the range of cosecant is 24
25 Summary: Concept 2: Domain and Range of Inverse trig functions The inverse trig functions are To construct inverse functions, we must have a property that our original functions are Is Sin 1-1 or not? In order to make the inverse of sin we must restrict our domain in the original. Creating the inverse sine function Sine Domain - Range - Inverse sine or Domain - Range - 25
26 Is Cos 1-1 or not? In order to make the inverse of cos we must restrict our domain in the original. Creating the inverse cosine function Cosine Domain - Range - Inverse cosine or Domain - Range - Creating the inverse tangent function Is Tan 1-1 or not? Then we must restrict it s domain too! Tangent Domain - Range - Inverse tangent or Domain - Range - 26
27 Summary: Find each value of : a) in degrees b) in radians Concept 3: Calculating the Degree Measure of inverse Trig Functions For inverse trig evaluations, be sure to use the restricted range values! Part One: Find the value of. 1) = Arc cos 3 2 1 2) = Arc tan -1 3) =Arc sin 2 4) = Arc sec ( ) 5) = Arc cot 3 6) = Arc csc ( ) Part Two: Find the value of 7) Find to the nearest degree: = Arc cos (-.6). 8) Find to the nearest degree: = Arc sin ( ). Part Three: Find the value of These problems involve 2 steps of evaluation, so do the inner one first. For inverse trig evaluations, be sure to use the restricted range values! 9) Find the exact value: sin (Arc tan 1) 10) Find the exact value: sec 2 Arc sin 2 10) Find the exact value: csc (Arc cos ) 11) Find the exact value: cot (Arc tan ) 27
28 SUMMARY Example 1: Exit Ticket 1) 2) 28
29 10-4&5 Domain and Range/Inverse Trig Functions : Homework 29
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32 Day 5: Chapter 10 Section 6 - Cofunctions Warm - Up Refer to the triangle below: a) What is the relationship between m A and m B? A b) What is the cos A? What is the sine B? c) What do you notice about the cosine and sine of complements? 12 13 C 5 B Sine and Cosine are called cofunctions. Any trigonometric function of an acute angle is equal to the cofunction of its complement. cos = sin (90 o - ) sin = cos (90 o - ) tan = cot (90 o - ) cot = tan (90 o - ) sec = csc (90 o - ) csc = sec (90 o - ) Concept 1: Expressing trig functions in terms of its cofunction. Examples: 1. 32
33 2. If x and (x + 20) are the measures of two acute angles and sin x = cos (x + 20), find x. 3. a) Express cos 75 as a function of an acute angle whose measure is less than 45 o. b) Find, to four decimal places, the value of the function value found in a. 4. a) Express sin 285 as a function of an acute angle whose measure is less than 45 o. b) Find, to four decimal places, the value of the function value found in a. 33
You try it! In 4-7, a) Rewrite each function value in terms of its cofunction. b) Find, to four decimal places, the value of the function value found in a. 4. sin 80 o 5. sec 83 o 6. csc 58 o 7. tan 172 o 34 In 8-11, the equation contains the measures of two acute angles. Find the value of θ for which the statement is true. 8. sin 10 o = cos θ 9. sin θ = cos 2θ 10. sec θ = csc (θ + 60 o ) 11. tan (θ + 5) = cot (2θ 20 o ) In 12-13, select the letter preceding the expression that best completes the sentence. 12. If θ is the measure of an acute angle and cos θ = sin 60 o, then cos θ equals: (a) 30 o (b) 60 o (c) 3 (d) ½ 2 13. If x is the measure of an acute angle and sin (x + 15 o ) = cos 45 o, then sin x equals: (a) ½ (b) 2 2 (c) 3 2 (d) 30 o 34
35 Summary/Closure Exit Ticket 35