MATH 30-1Foundations of Trigonometry Module Four Assignment

Module / Unit 4 - Assignment Booklet

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Lesson 1: Radians and Degrees in Standard Position

1. Calculate the number of degrees for a measure of . Sketch this angle in standard position.

2. Calculate the number of radians for a measure of 210. Sketch this angle in standard position.

3. Describe the relationship between degrees and radians. Use a sketch or image to help you visually describe the relationship. Then explain how this relationship can be used to convert between degrees and radians.

4. An arc has a central angle of 57.3 in a circle with a diameter of 24 cm. Determine the length of the arc to the nearest whole centimetre.

5. Determine the number of radians through which the minute hand of a clock rotates between 9:00 a.m. and 11:20 a.m. Enter only the fraction in terms of .

LESSON 1 SUMMARY

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Lesson 2: Applications of Radians 1. Determine one positive and one negative coterminal angle for the following angles.

a. 420

b.

2. Write an expression for all of the angles that are coterminal with each of the following angles.

a. 315

b.

3. Identify the angles coterminal with an angle of 650 that satisfy 360 360.

4. Explain, using images and examples, the relationship that exists among arc length, radius, and radian measure in a circle.

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5. Herbert goes to the playground to play on the swings. The length of each swing is 1.75 m.

a. What is the distance travelled on one swing, from maximum height to the next maximum height, if the central angle of the swing is 1.4 rad? Give your answer to the nearest tenth of a metre.

b. Herbert decides to swing higher. If the distance travelled on one swing is 3.5 m, what would be the central angle in degrees? Give your answer to the nearest degree.

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LESSON 2 SUMMARY

Digital Vision/Thinkstock

In this lesson you looked at objects rotating over and over again. The time-lapse image shows stars rotating around and around, as seen from the North Pole.

You learned that coterminal angles have the same terminal arm in standard position. There are an infinite number of coterminal angles for any given angle θ. Coterminal angles can be expressed in general form as θ + (360°)n, n ∈ I or θ + 2πn, n ∈ I.

Arc length is related to the radius of the circle and the central angle in radians. You learned that you could use either proportion of the circumference or the formula a = θr to determine arc length.

You went around and around in circles in this lesson. In Lesson 3 you will continue to study some of the properties of circles and determine how this information can help you with trigonometry.

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Lesson 3: The Unit Circle1. The following points all lie on the unit circle. Determine each unknown value.

a. , quadrant 2

b. , quadrant 3

c. (1, y)

2. Point P() is at the intersection of the unit circle and the terminal arm of angle . Determine the coordinates of P for each of the following points.

a.

b.

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c.

3. Given that , determine the following.

a. In which quadrant does terminate? Explain how you determined the quadrant.

b. Determine the value of the central angle for .

c. Determine the values of the central angle for . How is this related to coterminal angles?

4. Imagine you have been asked to explain, using words and images, how the equation for the unit circle is derived. Direct this explanation to someone who has studied the Pythagorean Theorem but is not familiar with the unit circle.

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UNIT CIRCLE TEMPLATE

Fill out all the information on the unit circle.

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LESSON 3 SUMMARY

© 3d_kot/35917383/Fotolia

The equation of the unit circle is x2 + y2 = 1 with its centre at (0, 0) and radius of 1. This equation can be derived from the Pythagorean theorem. This equation can be used to determine if a point is on the unit circle or to determine the value of the x- or y-coordinate when given one of the values.

You looked at specific points on the unit circle that correspond to eight or twelve equal parts of the circle. You wrote the coordinates of the specific points as exact values. Patterns can be used to help determine these points on the unit circle. The unit circle can be used to determine specific rotational angles when given the coordinates of a point and vice versa.

Source: Pre-Calculus 12. Whitby, ON: McGraw-Hill Ryerson, 2011. Reproduced with permission.

Source: Pre-Calculus 12. Whitby, ON: McGraw-Hill Ryerson, 2011. Reproduced with permission.

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Lesson 4: Trigonometric Ratios

1. The point P lies at the intersection of the unit circle and the terminal arm of angle in standard position.

a. Draw a diagram to model the situation.

b. Determine the values for tan , sec , and csc . Express your answers in lowest terms.

2. The point P(4, 5) is on the terminal arm of an angle .

a. Draw a diagram showing in standard position.

b. Calculate the exact values for the six trigonometric ratios.

3. Determine the exact values of the other reciprocal trigonometric ratios if

.

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4. At the beginning of the lesson, the idea of a bicycle tire with a radius of 1 unit being used to determine the height of the marker after a rotation of a specific angle was presented. Determine the exact value for the following trigonometric ratios.

a.

b.

c.

d. The marker on the bicycle tire is a point on the unit circle. Determine the coordinates of

the marker if the angle of rotation is .

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Stockbyte/Thinkstock

5. Determine the approximate value of each trigonometric ratio. Make sure to round your answer to four decimal places. Justify the sign for each ratio.

a. cos 2.51

b. csc (125)

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LESSON 4 SUMMARY

In this lesson you started with points on the unit circle. At the beginning of the lesson, a reference was made to a bicycle tire rotating though an angle θ with a marker on the tire and a radius of 1 unit. The coordinates of the marker would equal the cosine and sine ratio for the angle θ. Points where the terminal arm of angle θ and the unit circle intersect can be defined as P(θ) = (cos θ, sin θ).

Three reciprocal trigonometric ratios are used in this course.

This means if

Then

You can use P(θ) = (cos θ, sin θ) to help determine the six trigonometric ratios. So, cos

θ = x-coordinate, sin θ = y-coordinate, and or

iStockphoto/Thinkstock

To determine the trigonometric ratios for a point not on the unit circle, use the coordinates x and y and determine r. When given an angle, you can determine the exact values for trigonometric ratios using the unit circle or reference angles. Your calculator can determine approximate values for trigonometric ratios in the correct mode.For a summary on the unit circle, you may choose to watch the video Unit Circle Definition of Trig Functions.

Source: Khan Academy( BY-NC-SA 3.0

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Lesson 5: Trigonometric Equations1. Go back to Focus at the beginning of Lesson 5. How is the racing circuit an analogy for the

concepts learned in the lesson?

2. Solve 4 sin x 5 4 sin x to the nearest thousandth over

a. the domain 0 x 360

b. the real numbers

3. Latoya determined two approximate solutions to the equation 3 sin2 x tan2 x 1 to be these: x 1.027 and x 2.281. Verify that her solutions are correct. Explain your reasoning.

4. Solve sec2 sec 2 using exact values over

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a. the domain 2 x 2

b. the real numbers

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5.

Herbert has written a solution to the equation . He correctly

factored the expression and split the expression into two equations as shown.

The remainder of Herbert’s solution follows.

This part of the solution is not correct. Determine the error(s) and write a correct solution.

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LESSON 5 SUMMARY

F1online/Thinkstock

In previous math courses you explored how to solve trigonometric ratios from 0° to 360°. In this lesson you solved equations using radians and reciprocal trigonometric ratios for all angles.

You also learned to solve more complex trigonometric equations. You used skills learned previously to isolate the trigonometric term (e.g., sin θ, cos θ, tan θ). You then used the unit circle to obtain exact values of the angle. Your calculator can also be used to give approximate values. Read questions carefully to know what is required.

A general solution gives all possible solutions for an equation. To determine a general solution, find the solutions in one positive rotation and then determine other solutions using coterminal angles. A general solution may be written in the form θ = ___ + ___ nπ, n ∈ I. More than one expression is often required to express a general solution. In Lesson 6 you will begin to explore visual representations of trigonometric functions.

In upcoming lessons you will learn to apply this new understanding to solve problems that involve cyclic behaviour, such as amusement park rides.

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Lesson 6: Graphing and Transforming Sine and Cosine Functions 1

1. Use the function to answer the following questions.a. What is the period of the function?

b. What is the amplitude of the function?

c. Make a sketch of the graph that includes at least three complete cycles.

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2. Examine the two figures below.

a. Complete the following table using the graph in this question.

Property Value for f(x)

minimum

maximum

amplitude

period

x-intercept(s)

y-intercept(s)

domain

range

b. Determine an equation of the form f(x) a cos (bx) for the function.

c. Describe how the function g(x) cos x can be transformed to give the function f(x) shown in the diagram.

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3. Add an appropriate x- and y-scale to the given graph so the graph represents the function y 4 sin (5x).

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SINE TABLE AND GRAPH TEMPLATEFill in the following chart with the corresponding y-coordinate for the sine function. Sketch the graph.

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Angle y-coordinate

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COSINE TABLE AND GRAPH TEMPLATEFill in the following chart with the corresponding y-coordinate for the cosine function. Sketch the graph.

Angle Cos

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2

LESSON 6 SUMMARY

There are many ways to visually represent trigonometric functions. One of the most useful representations is a graph on a Cartesian plane. The sine and cosine functions have the same shape, and both can be transformed in the same manner as other functions.

The amplitude of a sinusoidal function is half the distance between the minimum and maximum value of the graph. The value of the amplitude is equal to the parameter a

and can be found using

The period of a periodic function is the horizontal distance taken to complete one graph

cycle. The period is related to the parameter b by the equation where P is the period.

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In Lesson 7 you will explore two other characteristics of a sinusoidal graph in addition to amplitude and period. In Module 5 you will begin to explore modeling cyclic behavior, such as the height of a mark on a bicycle tire over time. Think of the shape this graph will take.

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Lesson 7: Graphing and Transforming Sine and Cosine Functions 2

1. On grid paper, graph over the domain . Explain your process.

2. On grid paper, graph over the domain 2 x 2. Explain your process.

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3. Use the partial graph of f(x) to complete the questions that follow. Assume f(x) is a sine function.

a. Determine the domain and range of the function.

b. Determine the amplitude, period, phase shift, and vertical displacement for the function.

c. Determine an equation, in standard form, for f(x).

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4. Recall the information in Focus about the lynx and the hare. Suppose a trigonometric function of the form y a cos [b(x c)] d was used to model the lynx population.

a. What information would each parameter (a, b, c, and d) provide about the lynx population?

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LESSON 7 SUMMARY

Although sine and cosine functions can be transformed in the same manner as other functions, the characteristics of their graphs are given the more specific names of amplitude, period, phase shift and vertical displacement. These names correspond to the parameters of a, b, c, and d for a sine or cosine function written in the form

or

Parameter Characteristic of Graph

a a = amplitude

b

c c = phase shift right if c is positive, left if c is negative

d d = midline value

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MODULE 4 – FOUNDATIONS OF TRIGONOMETRY

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 4 is now complete.

Once you have received your corrected work, review your instructor’s comments and prepare for your module four test.

Text: Pre-Calculus 12

Section 4.1: Pages: 175 to 179 #3a,e, 4a,e, 6b,c, 11a,g, 13a,b, 15

Section 4.2: Pages: 186 to 190 #4a,e, 5a,e, 12,13

Section 4.3: Pages: 201 to 205 #2a,g,i, 5c,d, 8, 11c,d

Section 4.4: Pages: 211 to 214 #3a,d, 5a,c,d,

Section 5.1: Pages: 233 to 237 #2, 9c, 11b,c,

Section 5.2: Pages: 250 to 255 #1e,f, 2e, 6b, 16