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Precalculus 7th

David Cohen, Theodore B. Lee, David Sklar

Chapter 6

An Introduction to Trigonometry

Educators


Problem 1

Use the definitions (as in Example 2) to eval. uate the six trigonometric functions of (a) $\theta$ and (b) $\beta .$ In cases in which a radical occurs in a denominator, rationalize the denominator.
(GRAPHS CANNOT COPY)

Abdul V.
Numerade Educator

Problem 2

Use the definitions (as in Example 2) to eval. uate the six trigonometric functions of (a) $\theta$ and (b) $\beta .$ In cases in which a radical occurs in a denominator, rationalize the denominator.
(GRAPHS CANNOT COPY)

Abdul V.
Numerade Educator

Problem 3

Use the definitions (as in Example 2) to eval. uate the six trigonometric functions of (a) $\theta$ and (b) $\beta .$ In cases in which a radical occurs in a denominator, rationalize the denominator.
(GRAPHS CANNOT COPY)

Abdul V.
Numerade Educator

Problem 4

Use the definitions (as in Example 2) to eval. uate the six trigonometric functions of (a) $\theta$ and (b) $\beta .$ In cases in which a radical occurs in a denominator, rationalize the denominator.
(GRAPHS CANNOT COPY)

Abdul V.
Numerade Educator

Problem 5

Suppose that $\triangle A B C$ is a right triangle with $\angle C=90^{\circ}$.
If $A C=3$ and $B C=2,$ find the following quantities.
(a) $\cos A, \sin A, \tan A$
(b) $\sec B, \csc B, \cot B$

Abdul V.
Numerade Educator

Problem 6

Suppose that $\triangle A B C$ is a right triangle with $\angle C=90^{\circ}$.
If $A C=6$ and $B C=2,$ find the following quantities.
(a) $\cos A, \sin A, \tan A$
(b) $\sec B, \csc B, \cot B$

Abdul V.
Numerade Educator

Problem 7

Suppose that $\triangle A B C$ is a right triangle with $\angle C=90^{\circ}$.
If $A B=13$ and $B C=5,$ compute the values of the six trigonometric functions of angle $B$.

Abdul V.
Numerade Educator

Problem 8

Suppose that $\triangle A B C$ is a right triangle with $\angle C=90^{\circ}$.
If $A B=3$ and $A C=1,$ compute the values of the six trigonometric functions of angle $A$.

Abdul V.
Numerade Educator

Problem 9

Suppose that $\triangle A B C$ is a right triangle with $\angle C=90^{\circ}$.
If $A C=1$ and $B C=3 / 4,$ compute each quantity.
(a) $\sin B, \cos A$
(b) $\sin A, \cos B$
(c) $(\tan A)(\tan B)$

Abdul V.
Numerade Educator

Problem 10

Suppose that $\triangle A B C$ is a right triangle with $\angle C=90^{\circ}$.
If $A C=B C=4,$ compute the following.
(a) $\sec A, \csc A, \cot A$
(b) $\sec B, \csc B, \cot B$
(c) $(\cot A)(\cot B)$

Abdul V.
Numerade Educator

Problem 11

Suppose that $\triangle A B C$ is a right triangle with $\angle C=90^{\circ}$.
If $A B=25$ and $A C=24,$ compute each of the required quantities.
(a) $\cos A, \sin A, \tan A$
(b) $\cos B, \sin B, \tan B$
(c) $(\tan A)(\tan B)$

Abdul V.
Numerade Educator

Problem 12

Suppose that $\triangle A B C$ is a right triangle with $\angle C=90^{\circ}$.
If $A B=1$ and $B C=\sqrt{3} / 2,$ compute the following.
(a) $\cos A, \sin B$
(b) $\tan A, \cot B$
(c) $\sec A, \csc B$

Abdul V.
Numerade Educator

Problem 13

Use a calculator to compute cos $\theta, \sin \theta$ and tan $\theta$ for the given value of $\theta .$ Round each result to three decimal places.
$$\boldsymbol{\theta}=\mathbf{6 5}^{\circ}$$

Abdul V.
Numerade Educator

Problem 14

Use a calculator to compute cos $\theta, \sin \theta$ and tan $\theta$ for the given value of $\theta .$ Round each result to three decimal places.
$$\theta=21^{\circ}$$

Abdul V.
Numerade Educator

Problem 15

Use a calculator to compute cos $\theta, \sin \theta$ and tan $\theta$ for the given value of $\theta .$ Round each result to three decimal places.
$$\theta=38.5^{\circ}$$

Abdul V.
Numerade Educator

Problem 16

Use a calculator to compute cos $\theta, \sin \theta$ and tan $\theta$ for the given value of $\theta .$ Round each result to three decimal places.
$$\theta=12.4^{\circ}$$

Abdul V.
Numerade Educator

Problem 17

Use a calculator to compute cos $\theta, \sin \theta$ and tan $\theta$ for the given value of $\theta .$ Round each result to three decimal places.
$$\theta=80.06^{\circ}$$

Abdul V.
Numerade Educator

Problem 18

Use a calculator to compute cos $\theta, \sin \theta$ and tan $\theta$ for the given value of $\theta .$ Round each result to three decimal places.
$$\theta=0.99^{\circ}$$

Abdul V.
Numerade Educator

Problem 19

Use a calculator to compute sec $\theta, \csc \theta$ and cot $\theta$ (as in Example 5 ) for the given value of $\theta .$ Round each result to three decimal places.
$$\boldsymbol{\theta}=2 \boldsymbol{0}^{\circ}$$

Abdul V.
Numerade Educator

Problem 20

Use a calculator to compute sec $\theta, \csc \theta$ and cot $\theta$ (as in Example 5 ) for the given value of $\theta .$ Round each result to three decimal places.
$$\boldsymbol{\theta}=40^{\circ}$$

Abdul V.
Numerade Educator

Problem 21

Use a calculator to compute sec $\theta, \csc \theta$ and cot $\theta$ (as in Example 5 ) for the given value of $\theta .$ Round each result to three decimal places.
$$\theta=17.5^{\circ}$$

Abdul V.
Numerade Educator

Problem 22

Use a calculator to compute sec $\theta, \csc \theta$ and cot $\theta$ (as in Example 5 ) for the given value of $\theta .$ Round each result to three decimal places.
$$\theta=18.5^{\circ}$$

Abdul V.
Numerade Educator

Problem 23

Use a calculator to compute sec $\theta, \csc \theta$ and cot $\theta$ (as in Example 5 ) for the given value of $\theta .$ Round each result to three decimal places.
$$\theta=1^{\circ}$$

Abdul V.
Numerade Educator

Problem 24

Use a calculator to compute sec $\theta, \csc \theta$ and cot $\theta$ (as in Example 5 ) for the given value of $\theta .$ Round each result to three decimal places.
$$\theta=89.9^{\circ}$$

Abdul V.
Numerade Educator

Problem 25

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\sin \theta=3 / 4$$

Abdul V.
Numerade Educator

Problem 26

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\sin \theta=2 / 5$$

Abdul V.
Numerade Educator

Problem 27

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\cos \beta=\sqrt{3} / 5$$

Abdul V.
Numerade Educator

Problem 28

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\cos \beta=\sqrt{7} / 3$$

Abdul V.
Numerade Educator

Problem 29

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\sin A=5 / 13$$

Abdul V.
Numerade Educator

Problem 30

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\cos A=8 / 17$$

Abdul V.
Numerade Educator

Problem 31

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\tan B=4 / 3$$

Abdul V.
Numerade Educator

Problem 32

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\tan B=5$$

Abdul V.
Numerade Educator

Problem 33

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\sec C=3 / 2$$

Abdul V.
Numerade Educator

Problem 34

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\csc C=\sqrt{5} / 2$$

Abdul V.
Numerade Educator

Problem 35

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\cot \alpha=\sqrt{3} / 3$$

Abdul V.
Numerade Educator

Problem 36

Use the given information to determine the values of the remaining five trigonometric functions. (Assume that all of the angles are acute. When radicals appear in a denominator, rationalize the denominator.
$$\text { cot } \alpha=\sqrt{3} / 2$$

Abdul V.
Numerade Educator

Problem 37

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$\cos 60^{\circ}=\cos ^{2} 30^{\circ}-\sin ^{2} 30^{\circ}$$

Abdul V.
Numerade Educator

Problem 38

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$\cos 60^{\circ}=1-2 \sin ^{2} 30^{\circ}$$

Abdul V.
Numerade Educator

Problem 39

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$\sin ^{2} 30^{\circ}+\sin ^{2} 45^{\circ}+\sin ^{2} 60^{\circ}=3 / 2$$

Abdul V.
Numerade Educator

Problem 40

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$\sin 30^{\circ} \cos 60^{\circ}+\cos 30^{\circ} \sin 60^{\circ}=1$$

Abdul V.
Numerade Educator

Problem 41

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$2 \sin 30^{\circ} \cos 30^{\circ}=\sin 60^{\circ}$$

Abdul V.
Numerade Educator

Problem 42

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$2 \sin 45^{\circ} \cos 45^{\circ}=1$$

Abdul V.
Numerade Educator

Problem 43

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$\sin 30^{\circ}=\sqrt{\left(1-\cos 60^{\circ}\right) / 2}$$

Abdul V.
Numerade Educator

Problem 44

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$\cos 30^{\circ}=\sqrt{\left(1+\cos 60^{\circ}\right) / 2}$$

Abdul V.
Numerade Educator

Problem 45

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$\tan 30^{\circ}=\frac{\sin 60^{\circ}}{1+\cos 60^{\circ}}$$

Abdul V.
Numerade Educator

Problem 46

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$\tan 30^{\circ}=\frac{1-\cos 60^{\circ}}{\sin 60^{\circ}}$$

Abdul V.
Numerade Educator

Problem 47

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$1+\tan ^{2} 45^{\circ}=\sec ^{2} 45^{\circ}$$

Abdul V.
Numerade Educator

Problem 48

Verify that each equation is correct by evaluating each side. Do not use a calculator. The purpose of Exercises $37-48$ is twofold. First, doing the problems will help you to review the values in Table 1 of this section. Second, the exercises serve as an algebra review.
$$1+\cot ^{2} 60^{\circ}=\csc ^{2} 60^{\circ}$$

Abdul V.
Numerade Educator

Problem 49

(a) Use a calculator to evaluate $\cos 30^{\circ}$ and $\cos 45^{\circ} .$ Give your answers to as many decimal places as your calculator allows.
(b) In Table 1 there are expressions for $\cos 30^{\circ}$ and $\cos 45^{\circ}$ Use your calculator to evaluate these expressions, and check to see that the results agree with those in part (a).

Abdul V.
Numerade Educator

Problem 50

(a) Use a calculator to evaluate $\tan 30^{\circ}$ and $\tan 60^{\circ} .$ Give your answers to as many decimal places as your calculator allows.
(b) In Table 1 there are expressions for $\tan 30^{\circ}$ and $\tan 60^{\circ}$ Use your calculator to evaluate these expressions, and check to see that the results agree with those in part (a).

Abdul V.
Numerade Educator

Problem 51

Refer to the following figure. In the figure, the arc is a portion of a circle with center $\bar{O}$ and radius $r$.
(FIGURE CANNOT COPY)
(a) Use the figure and the right-triangle definition of sine to explain (in complete sentences) why $\sin 20^{\circ}<\sin 40^{\circ}<\sin 60^{\circ}$
(b) Use a calculator to verify that $\sin 20^{\circ}<\sin 40^{\circ}<\sin 60^{\circ}$

Abdul V.
Numerade Educator

Problem 52

Refer to the following figure. In the figure, the arc is a portion of a circle with center $\bar{O}$ and radius $r$.
(FIGURE CANNOT COPY)
(a) Use the figure and the right-triangle definition of cosine to explain (in complete sentences) why $\cos 20^{\circ}>\cos 40^{\circ}>\cos 60^{\circ}$
(b) Use a calculator to verify that $\cos 20^{\circ}>\cos 40^{\circ}>\cos 60^{\circ}$

Abdul V.
Numerade Educator

Problem 53

Refer to the following figure. In each case, say which of the two given quantities is larger. If the quantities are equal, say so. In either case, be able to support your conclusion.
(FIGURE CANNOT COPY)
$$\begin{aligned}
&\text { (a) } \cos \theta, \cos \beta\\
&\text { (b) } \sec \theta, \sec \beta
\end{aligned}$$

Abdul V.
Numerade Educator

Problem 54

Refer to the following figure. In each case, say which of the two given quantities is larger. If the quantities are equal, say so. In either case, be able to support your conclusion.
(FIGURE CANNOT COPY)
$$\begin{aligned}
&\text { (a) } \tan \theta, \tan \beta\\
&\text { (b) } \cot \theta, \cot \beta
\end{aligned}$$

Abdul V.
Numerade Educator

Problem 55

The following figure shows a $30^{\circ}-60^{\circ}$ right triangle, $\triangle A B C$. Prove that $A C=2 A B$. Suggestion: Construct $\triangle D B C$ as shown, congruent to $\triangle A B C .$ Then note that $\triangle A D C$ is equilateral. Note: This exercise provides a proof of the theorem on page 427.
(FIGURE CANNOT COPY)

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Problem 56

This exercise shows how to obtain radical expressions for $\sin 15^{\circ}$ and $\cos 15^{\circ} .$ In the figure, assume that $A B=B D=2$.
(FIGURE CANNOT COPY)
(a) In the right triangle $B C D,$ note that $D C=1$ because $D C$ is opposite the $30^{\circ}$ angle and $B D=2 .$ Use the Pythagorean theorem to show that $B C=\sqrt{3}$
(b) Use the Pythagorean theorem to show that $A D=2 \sqrt{2}+\sqrt{3}$
(c) Show that the expression for $A D$ in part (b) is equal to $\sqrt{6}+\sqrt{2} .$ Hint: Two nonnegative quantities are equal if and only if their squares are equal.
(d) Explain why $\angle B A D=\angle B D A$
(e) According to a theorem from geometry, an exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles. Apply this to $\triangle A B D$ with exterior angle $D B C=30^{\circ},$ and show that $\angle B A D=15^{\circ}$
(f) Using the figure and the values that you have obtained for the lengths, conclude that
$\sin 15^{\circ}=\frac{1}{\sqrt{6}+\sqrt{2}} \quad \cos 15^{\circ}=\frac{2+\sqrt{3}}{\sqrt{6}+\sqrt{2}}$
(g) Rationalize the denominators in part (f) to obtain $\sin 15^{\circ}=\frac{\sqrt{6}-\sqrt{2}}{4} \quad \cos 15^{\circ}=\frac{\sqrt{6}+\sqrt{2}}{4}$

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Problem 57

If an angle $\theta$ is an integral multiple of $3^{\circ},$ then the real number $\sin \theta$ is either rational or expressible in terms of radicals. You've already seen examples of this with the angles $30^{\circ}, 45^{\circ},$ and $60^{\circ} .$ The accompanying table gives the values of $\sin \theta$ for some multiples of $3^{\circ} .$ Use your calculator to check the entries in the table. Remark: If $\theta$ is an integral number of degrees but not a multiple of $3^{\circ},$ then the real number $\sin \theta$ cannot be expressed in terms of radicals within the real-number system.
$$\begin{array}{ll}
\boldsymbol{\theta} & \sin \boldsymbol{\theta} \\
\hline 3^{\circ} & \frac{1}{16}[(\sqrt{6}+\sqrt{2})(\sqrt{5}-1)-2(\sqrt{3}-1) \sqrt{5+\sqrt{5}}] \\
6^{\circ} & \frac{1}{8}(\sqrt{30-6 \sqrt{5}}-\sqrt{5}-1) \\
9^{\circ} & \frac{1}{8}(\sqrt{10}+\sqrt{2}-2 \sqrt{5-\sqrt{5}}) \\
12^{\circ} & \frac{1}{8}(\sqrt{10+2 \sqrt{5}}-\sqrt{15}+\sqrt{3}) \\
15^{\circ} & \frac{1}{4}(\sqrt{6}-\sqrt{2}) \\
18^{\circ} & \frac{1}{4}(\sqrt{5}-1) \\
\hline
\end{array}$$

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Problem 58

(a) Use the expression for $\sin 18^{\circ}$ given in Exercise 57 to show that the number $\sin 18^{\circ}$ is a root of the quadratic equation $4 x^{2}+2 x-1=0$
(b) Use the expression for $\sin 15^{\circ}$ given in Exercise 57 to show that the number $\sin 15^{\circ}$ is a root of the equation $16 x^{4}-16 x^{2}+1=0$

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