Chapter 14

Given \(\cos \theta=-\frac{4}{5}\) and \(90^{\circ}<\theta<180^{\circ},\) find the exact value of each expression. $$ \cos \frac{\theta}{2} $$

Solve each equation for \(0 \leq \theta<2 \pi\). $$ 3 \cos \theta=2 $$

Mental Math Find the value of each trigonometric expression. $$ \cos 183^{\circ} \cos 93^{\circ}+\sin 183^{\circ} \sin 93^{\circ} $$

In \(\triangle A B C, \angle C\) is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth. \(b=12, c=15\)

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

Given \(\cos \theta=-\frac{4}{5}\) and \(90^{\circ}<\theta<180^{\circ},\) find the exact value of each expression. $$ \tan \frac{\theta}{2} $$

Solve each equation for \(0 \leq \theta<2 \pi\). $$ 3 \tan \theta-1=\tan \theta $$

Find each exact value. Use a sum or difference identity. $$ \cos 105^{\circ} $$

In \(\triangle A B C, \angle C\) is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth. \(a=8.1, b=6.2\)

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

Given \(\cos \theta=-\frac{4}{5}\) and \(90^{\circ}<\theta<180^{\circ},\) find the exact value of each expression. $$ \cot \frac{\theta}{2} $$

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

Find each exact value. Use a sum or difference identity. $$ \tan 105^{\circ} $$

In \(\triangle A B C, \angle C\) is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth. \(b=4.3, c=9.1\)

Simplify each trigonometric expression. $$ \csc \theta \cos \theta \tan \theta $$

Given \(\cos \theta=-\frac{15}{17}\) and \(180^{\circ}<\theta<270^{\circ}\) , find the exact value of each expression. $$ \sin \frac{\theta}{2} $$

Solve each equation for \(0 \leq \theta<2 \pi\). $$ 3 \tan \theta+5=0 $$

Find each exact value. Use a sum or difference identity. $$ \tan 15^{\circ} $$

In \(\triangle A B C, \angle C\) is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth. \(a=17, c=22\)

Simplify each trigonometric expression. $$ \tan \theta(\cot \theta+\tan \theta) $$

Given \(\cos \theta=-\frac{15}{17}\) and \(180^{\circ}<\theta<270^{\circ}\) , find the exact value of each expression. $$ \cos \frac{\theta}{2} $$

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

Find each exact value. Use a sum or difference identity. $$ \sin 75^{\circ} $$

An observer on the ground at point \(A\) watches a rocket ascend. The observer is 1200 ft from the launch point \(B\) . As the rocket rises, the distance \(d\) from the observer to the rocket increases. a. Write a model for \(m \angle A .\) b. Find \(m \angle A\) if \(d=1500\) ft. Round your answer to the nearest degree. c. Find \(m \angle A\) if \(d=2000\) ft. Round your answer to the nearest degree.

Simplify each trigonometric expression. $$ \sin ^{2} \theta+\cos ^{2} \theta+\tan ^{2} \theta $$

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

Find each exact value. Use a sum or difference identity. $$ \cos 75^{\circ} $$

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\sin \theta=\frac{3}{8}\)

Critical Thinking In \(\triangle A B C, a=10\) and \(b=15 .\) a. Does the triangle have a greater area when \(m \angle C=1^{\circ}\) or when \(m \angle C=50^{\circ} ?\) b. Does the triangle have a greater area when \(m \angle C=50^{\circ}\) or when \(m \angle C=179^{\circ} ?\) c. For what measure of \(\angle C\) does \(\triangle A B C\) have the greatest area? Explain.

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

Given \(\cos \theta=-\frac{15}{17}\) and \(180^{\circ}<\theta<270^{\circ}\) , find the exact value of each expression. $$ \tan \frac{\theta}{2} $$

Given \(\cos \theta=-\frac{15}{17}\) and \(180^{\circ}<\theta<270^{\circ}\) , find the exact value of each expression. $$ \sec \frac{\theta}{2} $$

Solve each equation for \(0 \leq \theta<2 \pi\). $$ (\cos \theta)(\cos \theta+1)=0 $$

Find each exact value. Use a sum or difference identity. $$ \tan 75^{\circ} $$

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\cos \theta=\frac{7}{20}\)

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

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

Solve each equation for \(0 \leq \theta<2 \pi\). $$ (\sin \theta-1)(\sin \theta+1)=0 $$

Find each exact value. Use a sum or difference identity. $$ \cos 135^{\circ} $$

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\cos \theta=\frac{1}{5}\)

Forestry A forest ranger in an observation tower sights a fire \(39^{\circ}\) east of north. A ranger in a tower 10 miles due east of the first tower sights the fire at \(42^{\circ}\) west of north. How far is the fire from each tower?

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

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

Find each exact value. Use a sum or difference identity. $$ \tan 135^{\circ} $$

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\tan \theta=\frac{24}{7}\)

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

\(\triangle R S T\) has a right angle at \(\angle T .\) Use identities to show that each equation is true. $$ \cos 2 R=\frac{s^{2}-r^{2}}{t^{2}} $$

Geometry One of the congruent sides of an isosceles triangle is 10 \(\mathrm{cm}\) long. One of the congruent angles has a measure of \(54^{\circ} .\) Find the perimeter of the triangle. Round your answer to the nearest centimeter.

Solve each equation for \(0 \leq \theta<2 \pi\). $$ 2 \sin ^{2} \theta-1=0 $$

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