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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

(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.

Numerade Educator

(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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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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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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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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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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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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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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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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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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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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

Zachary M.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

(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.

Numerade Educator

(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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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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Use the Pythagorean identities to simplify the given expressions.

$$\frac{\sin ^{2} t+\cos ^{2} t}{\tan ^{2} t+1}$$

Zachary M.

Numerade Educator

Use the Pythagorean identities to simplify the given expressions.

$$\frac{\sec ^{2} t-1}{\tan ^{2} t}$$

Zachary M.

Numerade Educator

Use the Pythagorean identities to simplify the given expressions.

$$\frac{\sec ^{2} \theta-\tan ^{2} \theta}{1+\cot ^{2} \theta}$$

Zachary M.

Numerade Educator

Use the Pythagorean identities to simplify the given expressions.

$$\frac{\csc ^{4} \theta-\cot ^{4} \theta}{\csc ^{2} \theta+\cot ^{2} \theta}$$

Zachary M.

Numerade Educator

Prove that the equations are identities.

$$\sin ^{2} t-\cos ^{2} t=\frac{1-\cot ^{2} t}{1+\cot ^{2} t}$$

Zachary M.

Numerade Educator

$$\sin ^{2} t-\cos ^{2} t=\frac{1-\cot ^{2} t}{1+\cot ^{2} t}$$

Zachary M.

Numerade Educator

Prove that the equations are identities.

$$\frac{1}{1+\sec s}+\frac{1}{1-\sec s}=-2 \cot ^{2} s$$

Zachary M.

Numerade Educator

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.

Numerade Educator

Prove that the equations are identities.

$$\frac{\sec s+\cot s \csc s}{\cos s}=\csc ^{2} s \sec ^{2} s$$

Zachary M.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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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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.

Numerade Educator

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.

Numerade Educator

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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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.

Numerade Educator

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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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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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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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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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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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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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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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?

Check back soon!

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}

Check back soon!

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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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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