Chapter 14

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\csc \theta=\frac{21}{12}\)

Geometry The sides of a triangle are 15 in., 17 in., and 16 in. long. The smallest angle has a measure of \(54^{\circ} .\) Find the measure of the largest angle. Round your answer to the nearest degree.

Simplify each trigonometric expression. $$ \cos \theta\left(1+\tan ^{2} \theta\right) $$

a. open-Ended Sketch a triangle. Specify three of its measures so that you can use the Law of Cosines to find the remaining measures. b. Solve for the remaining measures of the triangle.

Solve each equation for \(0 \leq \theta<2 \pi\). $$ \tan \theta=\tan ^{2} \theta $$

Find each exact value. Use a sum or difference identity. $$ \cos \left(-15^{\circ}\right) $$

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\sec \theta=\frac{16}{9}\)

Simplify each trigonometric expression. $$ \frac{\tan \theta}{\sec \theta-\cos \theta} $$

Find the area of \(\triangle A B C\) . Round your answer to the nearest tenth. $$ m \angle C=68^{\circ}, b=12.9, c=15.2 $$

Sports A softball diamond is a square that is 65 \(\mathrm{ft}\) on a side. The pitcher's mound is 46 \(\mathrm{ft}\) from home plate. How far is the pitcher from third base?

Solve each equation for \(0 \leq \theta<2 \pi\). $$ \sin ^{2} \theta+3 \sin \theta=0 $$

Find each exact value. Use a sum or difference identity. $$ \tan \left(-15^{\circ}\right) $$

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\cot \theta=\frac{5}{4}\)

Find the area of \(\triangle A B C\) . Round your answer to the nearest tenth. $$ m \angle A=52^{\circ}, a=9.71, c=9.33 $$

Simplify each trigonometric expression. $$ \sec \theta \cos \theta-\cos ^{2} \theta $$

Solve each equation for \(0 \leq \theta<2 \pi\). $$ \sin \theta=-\sin \theta \cos \theta $$

Find each exact value. Use a sum or difference identity. $$ \sin 225^{\circ} $$

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\sin \theta=0.35\)

Simplify each trigonometric expression. $$ \sin \theta \csc \theta-\cos ^{2} \theta $$

If \(\sin 2 A=\sin 2 B,\) must \(A=B ?\) Explain.

Geometry The lengths of the sides of a triangle are \(7.6 \mathrm{cm}, 8.2 \mathrm{cm},\) and 5.2 \(\mathrm{cm} .\) Find the measure of the largest angle.

Solve each equation for \(0 \leq \theta<2 \pi\). $$ 2 \sin ^{2} \theta-3 \sin \theta=2 $$

Find each exact value. Use a sum or difference identity. $$ \cos 240^{\circ} $$

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\csc \theta=5.2\)

Find the area of \(\triangle A B C\) . Round your answer to the nearest tenth. $$ m \angle B=87^{\circ}, a=10.1, c=9.8 $$

Simplify each trigonometric expression. $$ \csc \theta-\cos \theta \cot \theta $$

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \sin 2 \theta $$

Navigation A pilot is flying from city \(A\) to city \(B,\) which is 85 mi due north. After flying 20 \(\mathrm{mi}\) , the pilot must change course and fly \(10^{\circ}\) east of north to avoid a cloudbank. a. If the pilot remains on this course for \(20 \mathrm{mi},\) how far will the plane be from city \(\mathrm{B} ?\) b. How many degrees will the pilot have to turn to the left to fly directly to city B? How many degrees from due north is this course?

Physics Two students set up a spring experiment similar to the one in Example \(7 .\) In their experiment, a weight was released 4 \(\mathrm{cm}\) below the rest position. It rose to 4 \(\mathrm{cm}\) above the rest position and returned to 4 \(\mathrm{cm}\) below the rest position once every 4 seconds. The equation \(h=-4 \cos \left(\frac{\pi}{2} t\right)\) models the height above and below the rest position at \(t\) seconds. a. Solve the equation for \(t\) . b. Find the times at which the weight is first at a height of \(1 \mathrm{cm}, 2 \mathrm{cm},\) and 3 \(\mathrm{cm}\) above the rest position. c. Find the times at which the weight is at a height of \(1 \mathrm{cm}, 2 \mathrm{cm},\) and 3 \(\mathrm{cm}\) below the rest position for the second time.

Find each exact value. Use a sum or difference identity. $$ \sin 390^{\circ} $$

Simplify each trigonometric expression. $$ \cos \theta+\sin \theta \tan \theta $$

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \cos 2 \theta $$

Find each exact value. Use a sum or difference identity. $$ \cos \left(-300^{\circ}\right) $$

In \(\triangle A B C, \angle C\) is a right angle. Two measures are given. Find the remaining sides and angles. Round your answers to the nearest tenth. \(b=8, c=17\)

Simplify each trigonometric expression. $$ \sec \theta\left(1+\cot ^{2} \theta\right) $$

Find the area of \(\triangle A B C\) . Round your answer to the nearest tenth. $$ m \angle C=33^{\circ}, a=1.2, b=0.9 $$

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \tan 2 \theta $$

Find each exact value. Use a sum or difference identity. $$ \tan 390^{\circ} $$

In \(\triangle A B C, \angle C\) is a right angle. Two measures are given. Find the remaining sides and angles. Round your answers to the nearest tenth. \(a=7, b=10\)

Simplify each trigonometric expression. $$ \csc ^{2} \theta\left(1-\cos ^{2} \theta\right) $$

Verify each identity. $$ \sin (A-B)=\sin A \cos B-\cos A \sin B $$

In \(\triangle A B C, \angle C\) is a right angle. Two measures are given. Find the remaining sides and angles. Round your answers to the nearest tenth. \(m \angle A=52^{\circ}, c=10\)

Simplify each trigonometric expression. $$ \frac{\cos \theta \csc \theta}{\cot \theta} $$

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \csc 2 \theta $$

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \sin \frac{\theta}{2} $$

Solve each equation for \(0 \leq \theta<2 \pi\) $$ \sec \theta=2 $$

Verify each identity. $$ \tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B} $$

In \(\triangle A B C, \angle C\) is a right angle. Two measures are given. Find the remaining sides and angles. Round your answers to the nearest tenth. \(m \angle A=34.2^{\circ}, b=5.7\)

Simplify each trigonometric expression. $$ \frac{\sin ^{2} \theta \csc \theta \sec \theta}{\tan \theta} $$

In \(\triangle A B C, m \angle A=40^{\circ}\) and \(m \angle B=30^{\circ} .\) Find each value to the nearest tenth. Find \(A C\) for \(B C=10.5 \mathrm{m}\)