Unit 5. Algebra 2. Name:

Transcription

1 Unit 5 Algebra 2 Name:

2 12.1 Day 1: Trigonometric Functions in Right Triangles Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples Theta Opposite Side of an Angle Adjacent Side of an Angle Hypotenuse Sine Cosine Tangent Reciprocal Functions

3 Cosecent Secent Cotangent SOH-CAH-TOA Steps in finding Trigonometric Functions: SOH CAH TOA sin θ = opp hyp cos θ = adj hyp tan θ = opp adj csc θ = hyp opp sec θ = hyp adj cot θ = adj opp 1. Mark the angle you are looking at. 2. Label the sides in relation to the angle you are looking at. (opposite, adjacent, hypotenuse) 3. Circle the sides you are supposed to use to make that trig function (use SOHCAHTOA) to help you. 4. Decide which side goes on top of the fraction (numerator) and which goes on bottom of the fraction (denominator). 5. Write the fraction. Example 1: Find Cos(A) Examples in Identifying Sides/and finding Trigonometric Values Example 2: Find Tan(B)

4 Example 1: In a right triangle, B is acute and cos B =. Find the value of tan B. Examples using Trigonometry Example 2: Find the values of the six trigonometric functions for angle θ. a. b. a) Given the triangle, find sec A b) Find ALL of the trigonometric values. You Try!

5 Relationships between Sides and Angles Example 3: In a right triangle, A and B are acute. Examples given NO triangle. a. If tan A =, what is cos A? b. If cos A =, what is tan A? c. If sin B =, what is tan B? Create your own steps given NO triangle and only information:

6 12.1 Day 2 Inverse Trigonometric Functions Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples Inverse Trigonometric Functions Notation of Inverse: Steps to find Theta given the side lengths. 1. Mark the angle that we are looking for, x. 2. Label the given sides in relation to x. 3. Decide which trig function goes with the sides given. 4. Write out the trig function. 5. Use your calculator to tell you the degree measure. Example 1: Find the measure of C. Round to the nearest tenth if necessary. Examples using Inverse Trig. Example 2: Find the measure of angle θ. a. b. c.

7 Example 3: Find the values of x and y using trigonometric functions a. Example 1: Find the missing angle, x You Try! 5 14 x Example 2: Find the missing angle, x 5 14 x Angle of Elevation Angle of Depression

8 Examples of Angle of Evaluation and Depression Example 1: The recommended angle of elevation for a ladder used in fire fighting is 75. At what height on a building does a 21-foot ladder reach if the recommended angle of elevation is used? Round to the nearest tenth. Example 2: The hill of the roller coaster has an angle of depression of 60. Its vertical drop is 195 feet. Estimate the length of the hill. Summary Activity When to use Trigonometric Functions? When to use Inverse Trigonometric Functions?

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10 12.2 : Angles and Angle Measures Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples Quadrants What s Positive and Negative? Standard Positive Initial Side Terminal Side

11 Degrees Axis Positive Rotation Negative Rotation Example 1: The terminal side of an angle of 81 degrees would be located in which quadrant? Then draw it. Examples of Angles: Example 2: The terminal side of an angle of 200 degrees would be located in which quadrant? Then draw it.

12 1: The terminal side of an angle of -20 degrees would be located in which quadrant? Then draw it. You Try! 2. The terminal side of an angle of -50 degrees would be located in which quadrant? Then draw it. Co-terminal Angles Examples of Co- Terminal Angles. Positive: Negative: Positive: Negative: Positive: Negative: Positive: Negative: You Try! 1. How do you find co-terminal angles? 2. Find a positive and negative co-terminal angle for Explain why 310 and 850 are NOT co-terminal angles.

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14 Algebra 2 Name: Review Sin θ Cscθ Cosθ Secθ Tanθ Cotθ 1. Evaluate the six trigonometric functions of the angle θ Solve ABC if C = 90 0 a = 7, and b = 12. Round answers to the tenths place. Define the following: Angle of depression: Angle of Elavation: 3. You are measuring the height of a flag pole. You stand 20 feet from the base of the pole. You measure the angle of elevation from a point on the ground to the top of the pole to be 65 o. Estimate the height of the pole to the nearest foot.

15 Draw the angle in standard position and find one positive and one negative coterminal angle o o Coterminals: Coterminals: Coterminals: Coterminals: Formula to convert degrees to radians: Formula to convert radians to degrees: Convert to radians o o o Convert to degrees Monster truck tires have a radius of 33 inches. How far does a monster truck travel in feet after just three fourths of a tire rotation?

16 12.3 Day 1 : Trigonometric Functions Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples sin θ = csc θ = Given a general point, find the six Trigonometric Functions cos θ = sec θ = tan θ = cot θ = Example 1: The terminal side of θ in standard position contains the point at (-3, -4). Find the exact values of the six trigonometric functions of θ. sin θ = csc θ = Given a point, find the exact values of the six trigonometric functions of theta. cos θ = sec θ = tan θ = cot θ = Example 2: Find the exact values of three trigonometric functions of θ if the terminal side of θ contains the point (5, -12). sin θ = csc θ = cos θ = sec θ = tan θ = cot θ =

17 The terminal side of θ in standard position contains each point. Draw the triangle, find x, y and r. Find the exact values of the six trigonometric functions of θ. 1. (6, 8) 2. ( 20, 21) You Try! sin θ = csc θ = sin θ = csc θ = cos θ = sec θ = cos θ = sec θ = tan θ = cot θ = tan θ = cot θ = Reference Angle Quadrant I Quadrant II Quadrant III Quadrant IV Positions of theta and reference angle: a. 210 b. 545 Examples of Reference Angles: c. d.

18 Sketch each angle. If a radian, convert to degree. If degree, convert to radian. Then find its reference angle. Lastly, find a positive and negative coterminal angle You Try! Degree: Reference Angle: Positive Coterminal Angle: Negative Coterminal Angle: Radian: Reference Angle: Positive Coterminal Angle: Negative Coterminal Angle: 3. Degree: Reference Angle: Positive Coterminal Angle: Negative Coterminal Angle:

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20 12.3 Day 2: Discovering the Unit Circle PONDER: Why is the circle above called a unit circle? Recall that the circumference of a circle is 2 r. Since this is a unit circle, what is the circumference? In the above diagram, label each quadrant, I through IV. Continue and indicate the signs of the x- and y-coordinates in each of the four quadrants above. UNIT CIRCLE BREAK DOWN The diagram to the right is a square. Let the diagonal be a length of First find the lengths of the sides. 2. What are the measures of the interior angles?

21 3. What type of triangles are these two triangles? Now, notice that the square we were just working with is inside the unit circle in quadrant I. Based on the sides of the square that we found, use that information to label the coordinate of point U. Use the coordinate of point U to label the other three angle markings. So, in each quadrant, the reference angles of 45 have coordinates of,. The signs change depending on what quadrant you are in. The diagram to the right is an equilateral triangle. Let each side have a length of Find the length of AD and DB. A 2. Find the length ofcd. C D 3. What are the measures of the interior angles? B

22 What type of triangles do we have? Notice that the equilateral triangle we were just working with is now inside the unit circle in quadrants I and IV. Based on the sides of the triangle that we found, use that information to label the coordinate of point A. Use a similar strategy to label the coordinate of point B. Follow the pattern and label the other two angle markings. So, in each quadrant, the reference angles of 30 have coordinates of,. The signs change depending on what quadrant you are in. Now, we are going to take our original equilateral triangle and break it up into the two separate triangles. Since our angle measures and our side lengths did not change, fill-in all the information we know about these two triangles on the diagram.

23 Notice that the separated equilateral triangle is now inside the unit circle in quadrants I and II. Based on the sides of the triangle, use that information to label the coordinate of point A and B. Follow the pattern and label the other two angle markings. So, in each quadrant, the reference angles of 60 have coordinates of,. The signs change depending on what quadrant you are in. adjacent opposite Recall that by definition, cos and sin. hypotenuse hypotenuse Looking at our 60 angle above in quadrant I, the adjacent piece is our x-value and the hypotenuse is our radius, or r. Still looking at our 60 angle in quadrant I, the opposite piece is our y-value. x y Then, cos and sin by substitution. Since this is a unit circle, r 1. This leads to the conclusion that r r cos x and sin y. Therefore, we can give an ordered pair for the x- and y-coordinates as cos,sin instead of ( xy, ). The next step is to put all of our information together onto one big unit circle.

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26 12.3 Day 3 : Trigonometric Functions Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples Hand Jive Learning it Steps for finding the Exact Values: 1. Identify the reference angle. (If a radian, turn into a degree first) **If it is above 360 degrees, find a co-terminal angle. 2. Identify the finger that will be folded down. 3. Do the hand jive! 4. Reduce if possible. 1. sin (135 ) 2. cos cos ( ) Example using the Hand Jive 4. sin ( ) 5. cos ( ) 6. sin ( ) Complete each of the following tables.

27 You Try! Radian Measure Degree Measure Sin Cos Radian Measure Degree Measure Sin Cos Rad Deg Ref. Angle sin cos tan cot sec csc

28 12.6 Day 1 : Circular Functions Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples Sine and Cosine If the side of an angle θ in standard position intersects the unit circle at, then x= and y=. Therefore, the coordinates of P can be written as P(, ). Example 1: The terminal side of angle θ in standard position intersects the unit Examples of finding Cos ɵ and Sin ɵ circle at ( ). Find cos θ and sin θ. Example 2: The terminal side of angle θ in standard position intersects the unit circle at each point P. Find cos θ and sin θ. 1. ( ) The terminal side of angle θ in standard position intersects the unit circle at each point P. Find cos θ and sin θ. 1. P(0, 1) You Try! 2. ( ) 3. ( )

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30 Unit Circle Quiz Review 1. Given the reference angle, provide the given information. Reference Angle Cos θ Sin θ Provide the given appropriate sign (positive or negative) about the given quadrants. Sign of Cos θ Sign of Sin θ Sign of Tan θ Quadrant I Quadrant II Quadrant III Quadrant IV 3. Write the reciprocal trigonometric functions using only sinθ cosθ and tanθ. Sec θ Csc θ Cot θ 3. Evaluate the function. Find the exact value of each trigonometric function. 1. tan cot cot ( 90 ) 4. cos tan 6. csc ( ) 7. cot 2π 8. tan

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32 12.6 Day 2 : Periodic Functions Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples Periodic Function Period Frequency Example 1: Determine the period and frequency of the function. Examples of periodic Functions : Example 2: Determine the period and frequency of the function. Example 3: Determine the period and frequency of the function.

33 You Try! Determine the period and frequency of each function cos sin sin 330 Review: 10. cos cos ( 60 ) 12. sin ( 390 ) 13. sin 5π 14. cos 3π 15. sin 16. sin 17. cos ( ) 18. cos ( )

34 12.7 Day 1 : Graphing Trigonometric Functions Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples Values Sin θ 0 or 0 90 or 180 or 270 or 360 or Values Cos θ 0 or 0 90 or 180 or 270 or 360 or Trigonometric functions can be graphed on the. Graphs of periodic functions have repeating patterns, or the horizontal length of each cycle is the. The of the graph of a sine or cosine function equals half the difference between the and values of the function. Tangent is a trigonometric function that has when graphed. As with other functions, trigonometric functions can be transformed. For the graphs of and, the amplitude = and the period =. Amplitude Period

35 Sine & Cosine Functions Parent Function Domain Range Amplitude Period y = sin θ y = cos θ Example 1: y = 4 cos Find the amplitude and period of each function: Example 2: y = 2 cos θ Example 3: y = 4 sin θ Example 4: y = sin 3θ Example 5: y = cos 2θ For the following examples, find the amplitude and period. 1. amplitude: period: You Try! 2. 3 amplitude: period: 3. amplitude: period: 4. amplitude: period:

36 For the following examples, find the amplitude and period based of the given graph. Then find a possible equation in the form of sin or cos for the function Given the graph, determine the amplitude and period: Amplitude: Period: Possible equation: Amplitude: Period: Possible equation: Amplitude: Period: Possible equation: Amplitude: Period: Possible equation: You Try! Find the amplitude, period and possible equation

37 12.7 Day 2 Trigonometric Function Word Problems Vocabulary, Main Topics, and Questions Definitions, Diagrams and Examples Example 1: Humans can hear sounds with a frequency of 40 Hz. Find the period of the function that models the sound waves. Let the amplitude equal 1 unit. Write a sine equation to represent the sound wave y as a function of time t. Example 2: The bass tuba can produce sounds with as low a frequency as 50 Hz. Find the period of the function that models the sound waves. Let the amplitude equal 1 unit. Determine the correct cosine equation to represent the sound wave y as a function of time t. Example 3: 10. The trombone can produce sounds with as low a frequency as 75 Hz. Find the period of the function that models the sound waves. Let the amplitude equal 2 unit. Determine the correct cosine equation to represent the sound wave y as a function of time t. Example 4: A mass on the end of a spring is at rest 80 cm above the ground. it is pulled down 60 cm and released at time t=0. It takes 2 seconds for the mass to return to the low position. determine the correct cosine to represent the wave of the spring y as a function of time t.

38 Unit 7 Multiple Choice Review Name 1

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41 Using the Pythagorean identity sin 2 θ + cos 2 θ 1 find cosθ if sinθ 1 4 and 180 < θ < 270?

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