Chapter 14

Verify each identity. $$ \csc \left(\theta-\frac{\pi}{2}\right)=-\sec \theta $$

From a hot-air balloon 3000 ft above the ground, you see a clearing whose angle of depression is \(20^{\circ} .\) Given that \(\sin 20^{\circ} \approx 0.34, \cos 20^{\circ} \approx 0.94\) and tan \(20^{\circ} \approx 0.36,\) find each distance to the nearest foot. a. your horizontal distance from the clearing b. your direct distance from the clearing

Verify each identity. $$\cos \theta \cot \theta=\frac{1}{\sin \theta}-\sin \theta$$

Use a double-angle identity to find the exact value of each expression. $$ \sin 240^{\circ} $$

Use the graph of the inverse of \(y=\sin \theta\) at the right. Find the measures of the angles whose sine is \(-1\)

Verify each identity. $$ \sec \left(\theta-\frac{\pi}{2}\right)=\csc \theta $$

Verify each identity. $$ \sin \theta \cot \theta=\cos \theta $$

Use a double-angle identity to find the exact value of each expression. $$ \cos 120^{\circ} $$

Use a double-angle identity to find the exact value of each expression. $$ \tan 120^{\circ} $$

Verify each identity. $$ \cot \left(\frac{\pi}{2}-\theta\right)=\tan \theta $$

In \(\triangle G H I, \angle H\) is a right angle, \(G H=40,\) and \(\cos G=\frac{40}{41} .\) Draw a diagram and find each value in fraction and in decimal form. a. \(\sin G\) b. \(\sin I\) c. \(\cot G\) d. \(\csc G\) e. \(\cos I\) f. \(\sec I\)

Verify each identity. $$ \cos \theta \tan \theta=\sin \theta $$

In \(\triangle D E F, m \angle E=54^{\circ}, d=14 \mathrm{ft},\) and \(f=20 \mathrm{ft} .\) Find \(e\)

Use a unit circle and \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles to find the degree measures of the angles. angles whose sine is \(\frac{1}{2}\)

Verify each identity. $$ \csc \left(\frac{\pi}{2}-\theta\right)=\sec \theta $$

A triangle has sides of lengths 10 \(\mathrm{cm}\) and \(16 \mathrm{cm},\) and the measure of the angle between them is \(130^{\circ} .\) Find the area of the triangle.

In \(\triangle P Q R, \angle R\) is a right angle and cot \(P=\frac{5}{12} .\) Draw a diagram. Find the values of the other five trigonometric functions of \(\angle P\) in fraction and in decimal form.

Verify each identity. $$ \sin \theta \sec \theta=\tan \theta $$

Use a double-angle identity to find the exact value of each expression. $$ \sin 90^{\circ} $$

Use a double-angle identity to find the exact value of each expression. $$ \cos 240^{\circ} $$

Use a unit circle and \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles to find the degree measures of the angles. angles whose tangent is \(\frac{\sqrt{3}}{3}\)

Verify each identity. $$ \tan \left(\theta-\frac{\pi}{2}\right)=-\cot \theta $$

Verify each identity. $$ \cos \theta \sec \theta=1 $$

Use a double-angle identity to find the exact value of each expression. $$ \tan 240^{\circ} $$

In \(\triangle A B C, m \angle B=52^{\circ}, a=15 \mathrm{in.},\) and \(c=10\) in. Find \(b\)

Use a unit circle and \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles to find the degree measures of the angles. angles whose sine is \(-\frac{1}{2}\)

Verify each identity. $$ \sec \left(\frac{\pi}{2}-\theta\right)=\csc \theta $$

Verify each identity. $$ \tan \theta \cot \theta=1 $$

Use a double-angle identity to find the exact value of each expression. $$ \cos 600^{\circ} $$

Use a unit circle and \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles to find the degree measures of the angles. angles whose tangent is \(-\sqrt{3}\)

Solve each trigonometric equation for \(0 \leq \theta < 2 \pi\) $$ \cos \left(\frac{\pi}{2}-\theta\right)=\csc \theta $$

Verify each identity. $$ \sin \theta \csc \theta=1 $$

Use a double-angle identity to find the exact value of each expression. $$ \sin 600^{\circ} $$

Use a calculator and inverse functions to find the radian measures of the angles. angles whose tangent is 1

Solve each trigonometric equation for \(0 \leq \theta < 2 \pi\) $$ \sin \left(\frac{\pi}{2}-\theta\right)=-\cos (-\theta) $$

In \(\triangle R S T, m \angle R=78^{\circ}, m \angle T=39^{\circ},\) and \(T S=19\) in. Find \(R S\)

In 1915 , the tallest flagpole in the world was in San Francisco. a. When the angle of elevation of the sun was \(55^{\circ}\) , the length of the shadow cast by this flagpole was 210 ft. Find the height of the flagpole to the nearest foot. b. What was the length of the shadow when the angle of elevation of the sun was \(34^{\circ} ?\) c. What do you need to assume about the flagpole and the shadow to solve these problems? Explain why.

Verify each identity. $$ \cot \theta=\frac{\csc \theta}{\sec \theta} $$

Use an angle sum identity to verify each identity. $$ \sin 2 \theta=2 \sin \theta \cos \theta $$

Use a calculator and inverse functions to find the radian measures of the angles. angles whose sine is 0.37

Solve each trigonometric equation for \(0 \leq \theta < 2 \pi\) $$ \tan \left(\frac{\pi}{2}-\theta\right)+\tan (-\theta)=0 $$

Find each angle measure to the nearest tenth of a degree. \(\cos ^{-1} \frac{\sqrt{2}}{2}\)

Simplify each trigonometric expression. $$ \tan \theta \cot \theta $$

Use an angle sum identity to verify each identity. $$ \tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta} $$

In \(\triangle D E F, d=15\) in, \(e=18\) in., and \(f=10\) in. Find \(m \angle F\)

Use a calculator and inverse functions to find the radian measures of the angles. angles whose sine is \((-0.78)\)

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

Find each angle measure to the nearest tenth of a degree. \(\tan ^{-1} 0.3333\)

Simplify each trigonometric expression. $$ 1-\cos ^{2} \theta $$

Use a half-angle identity to find the exact value of each expression. $$ \cos 15^{\circ} $$