Chapter 9

Liam said that if \(0<\theta<90,\) then when the degree measure of a fourth- quadrant angle is \(-\theta,\) the degree measure of the reference angle is \(\theta .\) Do you agree with Liam? Explain why or why not.

\(R\) is the point \((1,0), P^{\prime}\) is the point on a circle with center at the origin, \(O,\) and radius \(r,\) and \(m \angle R O P^{\prime}=\theta\) . Alicia said that the coordinates of \(P^{\prime}\) are \((r \cos \theta, r \sin \theta) .\) Do you agree with Alicia? Explain why or why not.

Explain why the calculator displays an error message when TAN 90 is entered.

Explain why sec \(\theta\) cannot equal 0.5

a. What are two possible measures of \(\theta\) if \(0^{\circ}<\theta<360^{\circ}\) and \(\sin \theta=\cos \theta ?\) Justify your answer. b. What are two possible measures of \(\theta\) if \(0^{\circ}<\theta<360^{\circ}\) and \(\tan \theta=1 ?\) Justify your answer.

If \(P\) is the point at which the terminal side of an angle in standard position intersects the unit circle, what are the largest and smallest values of the coordinates of \(P ?\) Justify your answer.

Is an angle of \(810^{\circ}\) a quadrantal angle? Explain why or why not.

In any right triangle, the acute angles are complementary. What is the relationship between the sine of the measure of an angle and the cosine of the measure of the complement of that angle? Justify your answer.

Sammy said that if a negative value is entered for \(\sin ^{-1}, \cos ^{-1},\) or \(\tan ^{-1}\) , the calculator will return a negative value for the measure of the angle. Do you agree with Sammy? Explain why or why not.

Hannah said that if \(\cos \theta=a,\) then \(\sin \theta=\pm \sqrt{1-a^{2}} .\) Do you agree with Hannah? Explain why or why not.

Explain why the calculator displays the same value for \(\sin 400^{\circ}\) as for \(\sin 40^{\circ} .\)

What is the value of \(\cos \theta\) when \(\tan \theta\) is undefined? Justify your answer.

Are the sine function and the cosine function one-to-one functions? Justify your answer.

Huey said that if the sum of the measures of two angles in standard position is a multiple of \(360,\) then the angles are coterminal. Do you agree with Huey? Explain why or why not.

Bebe said that if \(A\) is the measure of an acute angle of a right triangle, \(0<\sin A<1 .\) Do you agree with Bebe? Justify your answer.

In \(3-44,\) find the exact value. $$ \cos 30^{\circ} $$

In \(3-7,\) for each angle with the given degree measure: a. Draw the angle in standard position b. Draw its reference angle as an acute angle formed by the terminal side of the angle and the \(x\) -axis. c. Draw the reference angle in standard position. d. Give the measure of the reference angle. \(120^{\circ}\)

In \(3-38,\) find each function value to four decimal places. $$ \sin 28^{\circ} $$

In \(3-10,\) the terminal side of \(\angle R O P\) in standard position intersects the unit circle at \(P .\) If \(\mathrm{m} \angle R O P\) is \(\theta,\) find: a. \(\sin \theta\) b. \(\cos \theta\) c.tan \(\theta\) d. \(\sec \theta\) e. \(\csc \theta\) f. \(\cot \theta\) $$ P(0.6,0.8) $$

In \(3-11, P\) is the point at which the terminal side of an angle in standard position intersects the unit circle. The measure of the angle is \(\theta .\) For each point \(P\) the \(x\) -coordinate and the quadrant is given. Find: a. the \(y\) -coordinate of \(P\) b. \(\cos \theta\) c. \(\sin \theta\) d. \(\tan \theta\) \(\left(\frac{3}{5}, y\right),\) first quadrant

In \(3-7,\) draw each angle in standard position. $$ 45^{\circ} $$

The lengths of the sides of \(\triangle A B C\) are given. For each triangle, \(\angle C\) is the right angle and \(\mathrm{m} \angle A<\mathrm{m} \angle B .\) Find: a. \(\sin A\) b. \(\cos A\) c. \(\tan A\). \(6,8,10\)

In \(3-44,\) find the exact value. $$ \sin 30^{\circ} $$

In \(3-7,\) for each angle with the given degree measure: a. Draw the angle in standard position b. Draw its reference angle as an acute angle formed by the terminal side of the angle and the \(x\) -axis. c. Draw the reference angle in standard position. d. Give the measure of the reference angle. \(250^{\circ}\)

In \(3-38,\) find each function value to four decimal places. $$ \cos 35^{\circ} $$

In \(3-11, P\) is the point at which the terminal side of an angle in standard position intersects the unit circle. The measure of the angle is \(\theta .\) For each point \(P\) the \(x\) -coordinate and the quadrant is given. Find: a. the \(y\) -coordinate of \(P\) b. \(\cos \theta\) c. \(\sin \theta\) d. \(\tan \theta\) \(\left(\frac{5}{13}, y\right),\) fourth quadrant

In \(3-10\) , the terminal side of \(\angle R O P\) in standard position intersects the unit circle at \(P .\) If \(\mathrm{m} \angle R O P\) is \(\theta,\) find: a. \(\sin \theta\) b. \(\cos \theta\) c. the quadrant of \(\angle R O P\). $$ P(0.6,-0.8) $$

In \(3-7,\) draw each angle in standard position. $$ 540^{\circ} $$

The lengths of the sides of \(\triangle A B C\) are given. For each triangle, \(\angle C\) is the right angle and \(\mathrm{m} \angle A<\mathrm{m} \angle B .\) Find: a. \(\sin A\) b. \(\cos A\) c. \(\tan A\). \(5,12,13\)

In \(3-44,\) find the exact value. $$ \csc 30^{\circ} $$

In \(3-7,\) for each angle with the given degree measure: a. Draw the angle in standard position b. Draw its reference angle as an acute angle formed by the terminal side of the angle and the \(x\) -axis. c. Draw the reference angle in standard position. d. Give the measure of the reference angle. \(320^{\circ}\)

In \(3-38,\) find each function value to four decimal places. $$ \tan 78^{\circ} $$

In \(3-10,\) the terminal side of \(\angle R O P\) in standard position intersects the unit circle at \(P .\) If \(\mathrm{m} \angle R O P\) is \(\theta,\) find: a. \(\sin \theta\) b. \(\cos \theta\) c.tan \(\theta\) d. \(\sec \theta\) e. \(\csc \theta\) f. \(\cot \theta\) $$ P\left(-\frac{1}{6}, \frac{\sqrt{35}}{6}\right) $$

In \(3-11, P\) is the point at which the terminal side of an angle in standard position intersects the unit circle. The measure of the angle is \(\theta .\) For each point \(P\) the \(x\) -coordinate and the quadrant is given. Find: a. the \(y\) -coordinate of \(P\) b. \(\cos \theta\) c. \(\sin \theta\) d. \(\tan \theta\) \(\left(-\frac{6}{10}, y\right),\) second quadrant

In \(3-7,\) draw each angle in standard position. $$ -180^{\circ} $$

The lengths of the sides of \(\triangle A B C\) are given. For each triangle, \(\angle C\) is the right angle and \(\mathrm{m} \angle A<\mathrm{m} \angle B .\) Find: a. \(\sin A\) b. \(\cos A\) c. \(\tan A\). \(11,60,61\)

In \(3-44,\) find the exact value. $$ \tan 30^{\circ} $$

In \(3-7,\) for each angle with the given degree measure: a. Draw the angle in standard position b. Draw its reference angle as an acute angle formed by the terminal side of the angle and the \(x\) -axis. c. Draw the reference angle in standard position. d. Give the measure of the reference angle. \(-45^{\circ}\)

In \(3-38,\) find each function value to four decimal places. $$ \cos 100^{\circ} $$

In \(3-10,\) the terminal side of \(\angle R O P\) in standard position intersects the unit circle at \(P .\) If \(\mathrm{m} \angle R O P\) is \(\theta,\) find: a. \(\sin \theta\) b. \(\cos \theta\) c.tan \(\theta\) d. \(\sec \theta\) e. \(\csc \theta\) f. \(\cot \theta\) $$ P\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) $$

In \(3-11, P\) is the point at which the terminal side of an angle in standard position intersects the unit circle. The measure of the angle is \(\theta .\) For each point \(P\) the \(x\) -coordinate and the quadrant is given. Find: a. the \(y\) -coordinate of \(P\) b. \(\cos \theta\) c. \(\sin \theta\) d. \(\tan \theta\) \(\left(-\frac{1}{4}, y\right),\) third quadrant

In \(3-7,\) draw each angle in standard position. $$ -120^{\circ} $$

In \(3-44,\) find the exact value. $$ \cot 30^{\circ} $$

In \(3-7,\) for each angle with the given degree measure: a. Draw the angle in standard position b. Draw its reference angle as an acute angle formed by the terminal side of the angle and the \(x\) -axis. c. Draw the reference angle in standard position. d. Give the measure of the reference angle. \(405^{\circ}\)

In \(3-38,\) find each function value to four decimal places. $$ \sin 170^{\circ} $$

In \(3-10,\) the terminal side of \(\angle R O P\) in standard position intersects the unit circle at \(P .\) If \(\mathrm{m} \angle R O P\) is \(\theta,\) find: a. \(\sin \theta\) b. \(\cos \theta\) c.tan \(\theta\) d. \(\sec \theta\) e. \(\csc \theta\) f. \(\cot \theta\) $$ P\left(\frac{2 \sqrt{2}}{3}, \frac{1}{3}\right) $$

In \(3-7,\) draw each angle in standard position. $$ 110^{\circ} $$

The lengths of the sides of \(\triangle A B C\) are given. For each triangle, \(\angle C\) is the right angle and \(\mathrm{m} \angle A<\mathrm{m} \angle B .\) Find: a. \(\sin A\) b. \(\cos A\) c. \(\tan A\). \(16,30,34\)

In \(3-44,\) find the exact value. $$ \cos 60^{\circ} $$

In \(8-17,\) for each angle with the given degree measure, find the measure of the reference angle. \(100^{\circ}\)