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# Precalculus 7th

## Educators ### Problem 1

Evaluate each expression (as in Example 1 ).
(a) $\cos (11 \pi / 6)$
(b) $\cos (-11 \pi / 6)$
(c) $\sin (11 \pi / 6)$
(d) $\sin (-11 \pi / 6)$ Zachary M.

### Problem 2

Evaluate each expression (as in Example 1 ).
(a) $\cos (2 \pi / 3)$
(b) $\cos (-2 \pi / 3)$
(c) $\sin (2 \pi / 3)$
(d) $\sin (-2 \pi / 3)$ Zachary M.

### Problem 3

Evaluate each expression (as in Example 1 ).
(a) $\cos (\pi / 6)$
(b) $\cos (-\pi / 6)$
(c) $\sin (\pi / 6)$
(d) $\sin (-\pi / 6)$ Zachary M.

### Problem 4

Evaluate each expression (as in Example 1 ).
(a) $\cos (13 \pi / 4)$
(b) $\cos (-13 \pi / 4)$
(c) $\sin (13 \pi / 4)$
(d) $\sin (-13 \pi / 4)$ Zachary M.

### Problem 5

Evaluate each expression (as in Example 1 ).
(a) $\cos (5 \pi / 4)$
(b) $\cos (-5 \pi / 4)$
(c) $\sin (5 \pi / 4)$
(d) $\sin (-5 \pi / 4)$ Zachary M.

### Problem 6

Evaluate each expression (as in Example 1 ).
(a) $\cos (9 \pi / 4)$
(b) $\cos (-9 \pi / 4)$
(c) $\sin (9 \pi / 4)$
(d) $\sin (-9 \pi / 4)$ Zachary M.

### Problem 7

Evaluate each expression (as in Example 1 ).
(a) $\sec (5 \pi / 3)$
(b) $\csc (-5 \pi / 3)$
(c) $\tan (5 \pi / 3)$
(d) $\cot (-5 \pi / 3)$ Zachary M.

### Problem 8

Evaluate each expression (as in Example 1 ).
(a) $\sec (7 \pi / 4)$
(b) $\csc (-7 \pi / 4)$
(c) $\tan (7 \pi / 4)$
(d) $\cot (-7 \pi / 4)$ Zachary M.

### Problem 9

(a) List four positive real-number values of $t$ for which $\cos t=0$
(b) List four negative real-number values of $t$ for which $\cos t=0$ Zachary M.

### Problem 10

(a) List four positive real numbers $t$ such that $\sin t=1 / 2$
(b) List four positive real numbers $t$ such that $\sin t=-1 / 2$
(c) List four negative real numbers $t$ such that $\sin t=1 / 2$
(d) List four negative real numbers $t$ such that $\sin t=-1 / 2$ Zachary M.

### Problem 11

Use a calculator to evaluate the six trigonometric functions using the given real-number input. (Round the results to two decimal places.)
(a) 2.06
(b) $-2.06$ Zachary M.

### Problem 12

Use a calculator to evaluate the six trigonometric functions using the given real-number input. (Round the results to two decimal places.)
(a) 0.55
(b) $-0.55$ Zachary M.

### Problem 13

Use a calculator to evaluate the six trigonometric functions using the given real-number input. (Round the results to two decimal places.)
(a) $\pi / 6$
(b) $\pi / 6+2 \pi$ Zachary M.

### Problem 14

Use a calculator to evaluate the six trigonometric functions using the given real-number input. (Round the results to two decimal places.)
(a) 1000
(b) $1000-2 \pi$ Zachary M.

### Problem 15

Check that both sides of the identity are indeed equal for the given values of the variable $t$. For part ( $c$ ) of each problem, use your calculator.
$\sin ^{2} t+\cos ^{2} t=1$
(a) $t=\pi / 3$
(b) $t=5 \pi / 4$
(c) $t=-53$

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### Problem 16

Check that both sides of the identity are indeed equal for the given values of the variable $t$. For part ( $c$ ) of each problem, use your calculator.
$\tan ^{2} t+1=\sec ^{2} t$
(a) $t=3 \pi / 4$
(b) $t=-2 \pi / 3$
(c) $t=\sqrt{5}$

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### Problem 17

Check that both sides of the identity are indeed equal for the given values of the variable $t$. For part ( $c$ ) of each problem, use your calculator.
$\cot ^{2} t+1=\csc ^{2} t$
(a) $t=-\pi / 6$
(b) $t=7 \pi / 4$
(c) $t=0.12$

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### Problem 18

Check that both sides of the identity are indeed equal for the given values of the variable $t$. For part ( $c$ ) of each problem, use your calculator.
$\cos (-t)=\cos t$
(a) $t=\pi / 6$
(b) $t=-5 \pi / 3$
(c) $t=-4$

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### Problem 19

Check that both sides of the identity are indeed equal for the given values of the variable $t$. For part ( $c$ ) of each problem, use your calculator.
$\sin (-t)=-\sin t$
(a) $t=3 \pi / 2$
(b) $t=-5 \pi / 6$
(c) $t=13.24$

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### Problem 20

Check that both sides of the identity are indeed equal for the given values of the variable $t$. For part ( $c$ ) of each problem, use your calculator.
$\tan (-t)=-\tan t$
(a) $t=-4 \pi / 3$
(b) $t=\pi / 4$
(c) $t=1000$

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### Problem 21

Check that both sides of the identity are indeed equal for the given values of the variable $t$. For part ( $c$ ) of each problem, use your calculator.
$\sin (t+2 \pi)=\sin t$
(a) $t=5 \pi / 3$
(b) $t=-3 \pi / 2$
(c) $t=\sqrt{19}$

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### Problem 22

Check that both sides of the identity are indeed equal for the given values of the variable $t$. For part ( $c$ ) of each problem, use your calculator.
$\cos (t+2 \pi)=\cos t$
(a) $t=-5 \pi / 3$
(b) $t=\pi$
(c) $t=-\sqrt{3}$

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### Problem 23

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
$$\cos 2 t=2 \cos t ; t=\pi / 6$$ Zachary M.

### Problem 24

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
$$\sin 2 t=2 \sin t, t=\pi / 2$$ Zachary M.

### Problem 25

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
$$\text { If } \sin t=-3 / 5 \text { and } \pi<t<3 \pi / 2, \text { compute } \cos t \text { and } \tan t$$ Zachary M.

### Problem 26

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
$$\text { If } \cos t=5 / 13 \text { and } 3 \pi / 2<t<2 \pi, \text { compute } \sin t \text { and } \cot t$$ Zachary M.

### Problem 27

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
If $\sin t=\sqrt{3} / 4$ and $\pi / 2<t<\pi,$ compute tan $t$ Zachary M.

### Problem 28

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
$$\text { If } \sec s=-\sqrt{13} / 2 \text { and } \sin s>0, \text { compute } \tan s$$ Zachary M.

### Problem 29

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
\begin{aligned} &\text { If } \tan \alpha=12 / 5 \text { and } \cos \alpha>0, \text { compute } \sec \alpha, \cos \alpha, \text { and }\\ &\sin \alpha \end{aligned} Zachary M.

### Problem 30

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
$$\text { If } \cot \theta=-1 / \sqrt{3} \text { and } \cos \theta<0, \text { compute } \csc \theta \text { and } \sin \theta$$ Zachary M.

### Problem 31

In the expression $\sqrt{9-x^{2}},$ make the substitution $x=3 \sin \theta,$ where $0<\theta<\frac{\pi}{2},$ and show that the result is
$3 \cos \theta$ Zachary M.

### Problem 32

Make the substitution $u=2 \cos \theta$ in the expression $1 / \sqrt{4-u^{2}},$ and simplify the result. (Assume that $0<\theta<\pi .)$ Zachary M.

### Problem 33

In the expression $1 /\left(u^{2}-25\right)^{3 / 2},$ make the substitution $u=5 \sec \theta,$ where $0<\theta<\frac{\pi}{2},$ and show that the result is $\left(\cot ^{3} \theta\right) / 125$ Zachary M.

### Problem 34

In the expression $1 /\left(u^{2}-25\right)^{3 / 2},$ make the substitution $u=5 \sec \theta,$ where $0<\theta<\frac{\pi}{2},$ and show that the result is $\left(\cot ^{3} \theta\right) / 125$ Zachary M.

### Problem 35

In the expression $1 / \sqrt{u^{2}+7},$ let $u=\sqrt{7} \tan \theta,$ where $0<\theta<\pi / 2,$ and simplify the result. Zachary M.

### Problem 36

In the expression $\sqrt{x^{2}-a^{2}} / x(a>0),$ let $x=a \sec \theta,$ where $0<\theta<\frac{\pi}{2},$ and simplify the result. Zachary M.

### Problem 37

(a) If $\sin t=2 / 3,$ find $\sin (-t)$
(b) If $\sin \phi=-1 / 4,$ find $\sin (-\phi)$
(c) If $\cos \alpha=1 / 5,$ find $\cos (-\alpha)$
(d) If $\cos s=-1 / 5,$ find $\cos (-s)$ Zachary M.

### Problem 38

(a) If $\sin t=0.35,$ find $\sin (-t)$
(b) If $\sin \phi=-0.47,$ find $\sin (-\phi)$
(c) If $\cos \alpha=0.21,$ find $\cos (-\alpha)$
(d) If $\cos s=-0.56,$ find $\cos (-s)$ Zachary M.

### Problem 39

If $\cos t=-1 / 3,$ and $\frac{\pi}{2}<t<\pi,$ compute the following:
(a) $\sin (-t)+\cos (-t)$
(b) $\sin ^{2}(-t)+\cos ^{2}(-t)$ Zachary M.

### Problem 40

If $\sin (-s)=3 / 5,$ and $\frac{3 \pi}{2}<s<2 \pi,$ compute:
(a) $\sin s$
(c) $\cos s$
(b) $\cos (-s)$
(d) $\tan s+\tan (-s)$ Zachary M.

### Problem 41

Use one of the identities $\cos (t+2 \pi k)=$ $\cos t$ or $\sin (t+2 \pi k)=\sin t$ to evaluate each expression.
(a) $\cos \left(\frac{\pi}{4}+2 \pi\right)$
(b) $\sin \left(\frac{\pi}{3}+2 \pi\right)$
(c) $\sin \left(\frac{\pi}{2}-6 \pi\right)$ Zachary M.

### Problem 42

Use one of the identities $\cos (t+2 \pi k)=$ $\cos t$ or $\sin (t+2 \pi k)=\sin t$ to evaluate each expression.
(a) $\sin (17 \pi / 4)$
(b) $\sin (-17 \pi / 4)$
(c) $\cos 11 \pi$
(d) $\cos (53 \pi / 4)$
(e) $\tan (-7 \pi / 4)$
(f) $\cos (7 \pi / 4)$
(g) $\sec \left(\frac{11 \pi}{6}+2 \pi\right)$
(h) $\csc \left(2 \pi-\frac{\pi}{3}\right)$

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### Problem 43

Use the Pythagorean identities to simplify the given expressions.
$$\frac{\sin ^{2} t+\cos ^{2} t}{\tan ^{2} t+1}$$ Zachary M.

### Problem 44

Use the Pythagorean identities to simplify the given expressions.
$$\frac{\sec ^{2} t-1}{\tan ^{2} t}$$ Zachary M.

### Problem 45

Use the Pythagorean identities to simplify the given expressions.
$$\frac{\sec ^{2} \theta-\tan ^{2} \theta}{1+\cot ^{2} \theta}$$ Zachary M.

### Problem 46

Use the Pythagorean identities to simplify the given expressions.
$$\frac{\csc ^{4} \theta-\cot ^{4} \theta}{\csc ^{2} \theta+\cot ^{2} \theta}$$ Zachary M.

### Problem 47

Prove that the equations are identities.
$$\sin ^{2} t-\cos ^{2} t=\frac{1-\cot ^{2} t}{1+\cot ^{2} t}$$ Zachary M.

### Problem 48

Prove that the equations are identities.
$$\sin ^{2} t-\cos ^{2} t=\frac{1-\cot ^{2} t}{1+\cot ^{2} t}$$ Zachary M.

### Problem 49

Prove that the equations are identities.
$$\frac{1}{1+\sec s}+\frac{1}{1-\sec s}=-2 \cot ^{2} s$$ Zachary M.

### Problem 50

Prove that the equations are identities.
$$\frac{1+\tan s}{1-\tan s}=\frac{\sec ^{2} s+2 \tan s}{2-\sec ^{2} s}$$ Zachary M.

### Problem 51

Prove that the equations are identities.
$$\frac{\sec s+\cot s \csc s}{\cos s}=\csc ^{2} s \sec ^{2} s$$ Zachary M.

### Problem 52

Prove that the equations are identities.
$$(\tan \theta)\left(1-\cot ^{2} \theta\right)+(\cot \theta)\left(1-\tan ^{2} \theta\right)=0$$ Zachary M.

### Problem 53

Prove that the equations are identities.
$$\begin{array}{l} (\cos \alpha \cos \beta-\sin \alpha \sin \beta)(\cos \alpha \cos \beta+\sin \alpha \sin \beta)= \\ \cos ^{2} \alpha-\sin ^{2} \beta \end{array}$$ Zachary M.

### Problem 54

Prove that the equations are identities.
$$\cot \theta+\tan \theta+1=\frac{\cot \theta}{1-\tan \theta}+\frac{\tan \theta}{1-\cot \theta}$$

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### Problem 55

If $\sec t=13 / 5$ and $3 \pi / 2<t<2 \pi,$ evaluate $$\frac{2 \sin t-3 \cos t}{4 \sin t-9 \cos t}$$ Zachary M.

### Problem 56

If $\sec t=\left(b^{2}+1\right) /(2 b)$ and $\pi<t<3 \pi / 2,$ find $\tan t$ and
$\sin t .$ (Note: $b$ is negative. Why?) You should assume that $b<-1$ Zachary M.

### Problem 57

Use the accompanying figure to explain why the following four identities are valid. (The identities can be used to provide an algebraic foundation for the reference-angle technique that we've used to evaluate the trigonometric functions.) (FIGURE CANNOT COPY)
(i) $\sin (t+\pi)=-\sin t$
$\cos (t+\pi)=-\cos t$
(ii) $\sin (t-\pi)=-\sin t \quad$ (iv) $\cos (t-\pi)=-\cos t$

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### Problem 58

Use two of the results in Exercise 57 to verify the identity $\tan (t+\pi)=\tan t$ (You'll see the graphical aspect of this identity in Section $8.5 .$ ) Zachary M.

### Problem 59

In the equation $x^{4}+6 x^{2} y^{2}+y^{4}=32,$ make the substitutions
$$x=X \cos \frac{\pi}{4}-Y \sin \frac{\pi}{4} \quad \text { and } \quad y=X \sin \frac{\pi}{4}+Y \cos \frac{\pi}{4}$$
and show that the result simplifies to $X^{4}+Y^{4}=16$ (Hint: Evaluate the trigonometric functions, simplify the expressions for $x$ and $y,$ take out the common factor, and then substitute.)

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### Problem 60

Suppose that $\tan \theta=2$ and $0<\theta<\pi / 2$
(a) Compute $\sin \theta$ and $\cos \theta$
(b) Using the values obtained in part (a), make the substitutions
$$x=X \cos \theta-Y \sin \theta \quad \text { and } \quad y=X \sin \theta+Y \cos \theta$$
in the expression $7 x^{2}-8 x y+y^{2},$ and simplify the result.

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### Problem 61

In this exercise, we are going to find the minimum value of the function
$$f(t)=\tan ^{2} t+9 \cot ^{2} t \quad 0<t<\frac{\pi}{2}$$
(a) Set your calculator in the radian mode and complete the table. Round the values you obtain to two decimal places. \begin{tabular}{cccccccc}
$t$ & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\
\hline$f(t)$ & & & & & & & \\\hline\end{tabular}
(b) Of the seven outputs you calculated in part (a), which is the smallest? What is the corresponding input?
(c) Prove that $\tan ^{2} t+9 \cot ^{2} t=(\tan t-3 \cot t)^{2}+6$
(d) Use the identity in part (c) to explain why $\tan ^{2} t+9 \cot ^{2} t \geq 6$
(e) The inequality in part (d) tells us that $f(t)$ is never less than 6. Furthermore, in view of part (c), $f(t)$ will equal 6 when $\tan t-3 \cot t=0 .$ From this last equation, show that $\tan ^{2} t=3,$ and conclude that $t=\pi / 3 .$ In summary, the minimum value of $f$ is $6,$ and this occurs when $t=\pi / 3 .$ How do these values compare with your answers in part (b)?

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### Problem 62

Let $f(\theta)=\sin \theta \cos \theta\left(0 \leq \theta \leq \frac{\pi}{2}\right)$
(a) Set your calculator in the radian mode and complete the table. Round the results to two decimal places. \begin{tabular}{cccccccc} $\theta$ & 0 & $\frac{\pi}{10}$ & $\frac{\pi}{5}$ & $\frac{\pi}{4}$ & $\frac{3 \pi}{10}$ & $\frac{2 \pi}{5}$ & $\frac{\pi}{2}$ \\ \hline$f(\theta)$ & & & & & & \\\hline\end{tabular}
(b) What is the largest value of $f(\theta)$ in your table in part (a)?
(c) Show that $\sin \theta \cos \theta \leq 1 / 2$ for all real numbers $\theta$ in the interval $0 \leq \theta \leq \pi / 2 .$ Hint: Use the inequality $\sqrt{a b} \leq(a+b) / 2[\text { given in Exercise } 40(b)$ in Section $2.3],$ with $a=\sin \theta$ and $b=\cos \theta$
(d) Does the inequality $\sin \theta \cos \theta \leq \frac{1}{2}$ hold for all real numbers $\theta ?$

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### Problem 63

Consider the equation
$$2 \sin ^{2} t-\sin t=2 \sin t \cos t-\cos t$$
(a) Evaluate each side of the equation when $t=\pi / 6$
(b) Evaluate each side of the equation when $t=\pi / 4$
(c) Is the given equation an identity?

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### Problem 64

Suppose that
$$f(t)=(\sin t \cos t)(2 \sin t-1)(2 \cos t-1)(\tan t-1)$$
(a) Compute each of the following: $f(0), f(\pi / 6), f(\pi / 4)$ $f(\pi / 3),$ and $f(\pi / 2)$
(b) Is the equation $f(t)=0$ an identity?

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### Problem 65

In Section 7.1 we pointed out that one of the advantages in using radian measure is that many formulas then take on particularly simple forms. Another reason for using radian measure is that the trigonometric functions can be closely approximated by very simple polynomial functions. To see examples of this, complete the tables in Exercises $65-68 .$ Round (or, for exact values, simply report the answers to six decimal places. In Exercises 67 and $68,$ note that the higher-degree polynomial provides the better approximation. Note: The approximating polynomials are known as Taylor polynomials, after the English mathematician Brook Taylor $(1685-1731) .$ The theory of Taylor polynomials is developed in calculus.
\begin{aligned}&(a)\\&\begin{array}{lll}t & 1-\frac{1}{2} t^{2} & \cos t \\\hline 0.02 & & \\ 0.05 & & \\0.1 & & \\0.2 & & \\0.3 & & \\\hline\end{array}\end{aligned}
(b) On the same set of axes, graph the functions $1-\frac{1}{2} t^{2}$ and cos $t$. Use a window extending from $-\pi$ to $\pi$ in the $x$ -direction and $-1$ to 1 in the $y$ -direction. What do you observe?

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### Problem 66

In Section 7.1 we pointed out that one of the advantages in using radian measure is that many formulas then take on particularly simple forms. Another reason for using radian measure is that the trigonometric functions can be closely approximated by very simple polynomial functions. To see examples of this, complete the tables in Exercises $65-68 .$ Round (or, for exact values, simply report the answers to six decimal places. In Exercises 67 and $68,$ note that the higher-degree polynomial provides the better approximation. Note: The approximating polynomials are known as Taylor polynomials, after the English mathematician Brook Taylor $(1685-1731) .$ The theory of Taylor polynomials is developed in calculus.
$(a)$\begin{tabular}{lll}$t$ & $t-\frac{1}{6} t^{3}$ & $\sin t$ \\\hline 0.02 & & \\0.05 & & \\
0.1 & & \\0.2 & & \\0.3 & & \\\hline\end{tabular}
(b) On the same set of axes, graph the functions $t-\frac{1}{6} t^{3}$ and $\sin t$ Use a window extending from $-\pi$ to $\pi$ in the $x$ -direction and $-1$ to 1 in the $y$ -direction. What do you observe?

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### Problem 67

In Section 7.1 we pointed out that one of the advantages in using radian measure is that many formulas then take on particularly simple forms. Another reason for using radian measure is that the trigonometric functions can be closely approximated by very simple polynomial functions. To see examples of this, complete the tables in Exercises $65-68 .$ Round (or, for exact values, simply report the answers to six decimal places. In Exercises 67 and $68,$ note that the higher-degree polynomial provides the better approximation. Note: The approximating polynomials are known as Taylor polynomials, after the English mathematician Brook Taylor $(1685-1731) .$ The theory of Taylor polynomials is developed in calculus.
(a)\begin{tabular}{llll}$x$ & $\frac{1}{3} x^{3}+x$ & $\frac{2}{15} x^{5}+\frac{1}{3} x^{3}+x$ & $\tan x$ \\\hline 0.1 & & & \\0.2 & & & \\0.3 & & & \\0.4 & & & \\0.5 & & & \\\hline \end{tabular}

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### Problem 68

In Section 7.1 we pointed out that one of the advantages in using radian measure is that many formulas then take on particularly simple forms. Another reason for using radian measure is that the trigonometric functions can be closely approximated by very simple polynomial functions. To see examples of this, complete the tables in Exercises $65-68 .$ Round (or, for exact values, simply report the answers to six decimal places. In Exercises 67 and $68,$ note that the higher-degree polynomial provides the better approximation. Note: The approximating polynomials are known as Taylor polynomials, after the English mathematician Brook Taylor $(1685-1731) .$ The theory of Taylor polynomials is developed in calculus.
(a)\begin{tabular}{llll}$x$ & $x^{2}+x$ & $\frac{1}{3} x^{3}+x^{2}+x$ & $e^{x} \sin x$ \\
\hline 0.1 & & \\0.2 & & \\0.3 & & \\0.4 & & \\0.5 & & \\\hline\end{tabular}
(b) On the same set of axes, graph the functions $x^{2}+x$ $\frac{1}{3} x^{3}+x^{2}+x,$ and $e^{x} \sin x .$ Use a window extending from $-\pi$ to $\pi$ in the $x$ -direction and $-10$ to 10 in the $y$ -direction. What do you observe?

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### Problem 69

Figure A on the following page shows two $x$ -y coordinate systems. (The same unit of length is used on all four axes.) In the coordinate system on the left, the curve is a portion of the unit circle
$$x^{2}+y^{2}=1$$
and $A$ is the point $(1,0) .$ The points $B, C, D, E,$ and $F$ are located on the circle according to the information in the following table.
$$\begin{array}{llllll}\hline \text { Arc } & \overline{A B} & \overline{A C} & \overline{A D} & \overline{A E} & A F \\\text { Length } & \frac{\pi}{12} & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{5 \pi}{12} \\\hline\end{array}$$
Determine the $y$ -coordinates of the points $P, Q, R, S,$ and $T$ Give an exact expression for each answer and, where appropriate, a calculator approximation rounded to three decimal places.

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