MATH 30-1Trigonometry of Applications & Identities Assignment Module Five

Module / Unit 5 - Assignment Booklet

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Lesson 1: The Tangent Function1. Determine tan and in the diagram to the nearest hundredth.

2. A typical drag strip is mi long or 402 m. Suppose a set of bleachers 3 m deep are set up 20 m away from the track.

a. Which position in the bleachers requires you to turn your head farthest away from a straight-forward position to watch the race? What is the greatest angle required for this seat?

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b. Which position in the bleachers requires you to turn your head the smallest angle from a straight-forward position to watch the race? What is the greatest angle required for this seat?

c. Suppose you are sitting in the front middle seat. Write an equation that represents distance, d, of the track visible by turning your head an angle of in one direction. Include the domain for this scenario.

d. Sketch the graph of the equation you determined in part c. over the domain you gave.

e. Determine the maximum and minimum values of d and .

LESSON 1 SUMMARY

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In this lesson you explored the tangent function. The tangent function can be defined as the y-coordinate of the intersection of the terminal arm and a vertical line tangent to the right side of the unit circle as shown in the diagram.

When the terminal arm is in quadrant 1 or quadrant 4, the intersection of the terminal arm is used. When the terminal arm is in quadrant 2 or quadrant 3, an extension of

the terminal arm is used.

Although y = tan x is periodic, it is very different from y = sin x or y = cos x. The function y = tan x is not defined for all x values, and its graph contains vertical asymptotes and has no minimum or maximum value.

sin θ, cos θ, and tan θ are related by the equation and the slope of the terminal arm in standard position at angle θ is equal to tan θ.

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Lesson 2: Equations and Graphs of Trigonometric Functions1. The height of the two pistons, from the beginning of Lesson 2, can be modelled using the

equations and for the piston with the waving rod and the piston with the rigid rod, respectively.

a. The mechanism with the waving rod is what is found in the engines of most cars. Why might someone choose to use the equation instead of

to model piston movement? Would this choice always be appropriate?

b. What information can you accurately determine using instead of

? What information is not completely accurate?

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2. The path of a swing could be modelled by the function , where h is the height in centimetres above the ground and t is the time in seconds.

a. What is the maximum height of the swing?

b. How many seconds does it take to reach maximum height after the person starts?

c. What is the minimum height of the swing?

d. How many seconds does it take to reach minimum height?

e. For how many seconds within one cycle is the swing less than 60 cm above the ground? Round your answer to the nearest tenth of a second.

f. Determine the height of the swing after 10 s have passed.

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3. Windsor, Ontario, is located at latitude 42N. The table shows the number of hours of daylight in Windsor on the 21st day of each month and the day of the year on which it occurs.

Hours of Daylight by Day of the Year for Windsor, Ontario

21 52 80 111 141 172 202 233 264 294 325 355

9.62 10.87 12.20 13.64 14.79 15.25 14.81 13.64 12.22 10.82 9.59 9.08

a. Draw a scatter plot for the number of hours of daylight, h, in Windsor on the day of the year, t.

b. Write the sinusoidal function that models the number of hours of daylight.

c. Graph the function from question b.

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d. Estimate the number of hours of daylight on July 24 (day 205).

e. Estimate the days of the year that would have 14 hours of daylight.

LESSON 2 SUMMARY

In this lesson you studied how trigonometric models can be used to solve problems. A mathematical model is a representation of a system using mathematical ideas and language. It can take on many forms, such as graphs or equations, as was demonstrated in this lesson. Most models are only an approximation of reality. Understanding the limitations of a model can help you decide how accurate a model's predictions will be.

Lesson 3: Trigonometric Identities

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1. Determine the non-permissible values for the expression .

2. Consider the identity .

a. Verify the identity for the values and 150.

b. Verify the identity graphically.

c. Suppose the identity were used to simplify an expression to tan x. For what values is this new expression defined?

3. Simplify each of the following expressions to a single trigonometric function. Be sure to state any non-permissible values.

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a.

b.

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LESSON 3 SUMMARY

In Focus you read about two race cars that were identical in every way other than colour. This analogy was used to describe trigonometric identities, equations with sides that look different but generally act the same. An identity is a mathematical equation that is true for any permissible value. In this lesson you worked with Pythagorean and reciprocal identities as well as some other less common identities. It is possible to verify an identity both graphically and numerically when given a potential identity.

Identities can be used to simplify some expressions; if one side of the identity appears in an expression, the other side of the identity can replace it. When simplifying using an identity, any restrictions on the identity also apply to the simplified expression.

In the next lesson you will learn some new trigonometric identities and use identities to help solve trigonometric equations.

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Lesson 4: Using Identities to Solve Equations1. Simplify sin 4x cos 3x cos 4x sin 3x to a single trigonometric function.

2. Determine the exact value of using a sum, difference, or double-angle identity.

3. Determine the general solution to the equation

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LESSON 4 SUMMARY

There is no limit to the number of possible trigonometric identities. In this lesson you focused on a set of related identities: the sum, difference, and double-angle identities. You also saw that it is often possible to predict an identity from previous knowledge.

Sum Identities Difference Identities

sin (A + B) = sin A cos B + cos A sin B sin (A − B) = sin A cos B − cos A sin B

cos (A + B) = cos A cos B − sin A sin B cos (A − B) = cos A cos B + sin A sin B

Double-Angle Identities

sin 2A = 2 sin A cos A cos 2A = cos2A − sin2A

cos 2A = 2 cos2A − 1

cos 2A = 1 − 2 sin2A

Identities are useful when you are interested in changing the form of an equation or expression; you simply exchange one side of the identity for the other. Remember, if your identity has a restricted domain, that restriction will carry on to the equation or expression if the identity is used.

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Lesson 5: Proving Trigonometric Identities1. a. Why is it necessary to prove a trigonometric identity before using the identity?

b. Explain the metaphor “an identity is the specifications of a car, the proof is the design.”

2. If the identity is used in the proof of another identity, for what values of x is the proof valid?

3. Prove each identity. Show your steps and/or explain your reasoning.

a. b.

c.

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LESSON 5 SUMMARY

In this lesson you learned about mathematical proofs, which are at the core of mathematics. A proof is just an argument showing that a statement must be true. Once you know the statement is always true, you can use the statement in future scenarios, such as simplifying expressions and solving other equations.

For the type of proof you learned in this lesson, the goal is to show that both sides of the identity are equal by manipulating each side independently until they are the same expression. There is no specific pattern for how to proceed with a given proof. A bit of luck, determination, and some intuition will help you complete a proof.

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MODULE 5 - TRIGONOMETRY OF APPLICATIONS & IDENTITIES SUMMATIVE ASSIGNMENTComplete the following questions from your text book. Show steps completely and clearly, as marks are assigned for mathematical literacy and communication. Always use graph paper, rulers, and pencils as necessary. Attach securely to this booklet before you hand everything in.

Module 5 is now complete. Once you have received your corrected work, review your instructor’s comments and prepare for your module five test.

Text: Pre-Calculus 12 – Chapters 5 and 6

Chapter 5 Trigonometric Functions and Graphs

Section 5.1: Pages 233 to 237 #9, 11, 12, 14

Section 5.2; Pages 250 to 255 #2, 6, 14

Section 5.3: Pages 262 to 265 #2, 5, 8, 9

Section 5.4: Pages 275 to 281 #4a,b, 5all, 12, 19

Chapter 6 Trigonometric Identities

Section 6.1: Pages 296 to 298 #4, 5, 8, 15

Section 6.2: Pages 306 to 308 #4, 5, 20

Section 6.3: Pages 314 to 315 #2a,b, 3a,b, 10a, 11a,b, 13

Section 6.4: Pages 320 to 321 #3, 6, 8, 11, 16