Chapter 14

Find each angle measure to the nearest tenth of a degree. $$ \tan ^{-1} \sqrt{2} $$

Verify each identity. $$ 1+\sec \theta=\frac{1+\cos \theta}{\cos \theta} $$

Use half-angle identities to write each expression, using trigonometric functions of \(\theta\) instead of \(\frac{\theta}{4} .\) $$ \tan \frac{\theta}{4} $$

Use the sum and difference formulas to verify each identity. $$ \cos (\pi+\theta)=-\cos \theta $$

Error Analysis A student solved an equation as shown below. What error did the student make? $$\begin{aligned} \theta &=\cos ^{-1} 0.5 \\ &=\frac{1}{\cos 0.5} \\ & \approx \frac{1}{0.88} \\ & \approx 1.14 \end{aligned}$$

In \(\triangle A B C, \angle B\) is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth. $$ b=5, c=4 $$

Verify each identity. $$ \frac{1+\tan \theta}{\tan \theta}=\cot \theta+1 $$

Use the Tangent Half-Angle Identity and a Pythagorean identity to prove each identity. a. \(\tan \frac{A}{2}=\frac{\sin A}{1+\cos A}\) b. \(\tan \frac{A}{2}=\frac{1-\cos A}{\sin A}\)

In \(\triangle X Y Z, \angle Z\) is a right angle and \(\tan X=\frac{8}{15}\) What is \(\sin Y ?\) F. \(\frac{8}{17}\) G. \(\frac{15}{17}\) H. \(\frac{17}{15}\) J. \(\frac{15}{8}\)

Reasoning For any parallelogram, prove that the sum of the squares of the lengths of the diagonals equals twice the sum of the squares of the lengths of two adjacent sides.

Verify each identity. $$ \frac{\cot \theta \sin \theta}{\sec \theta}+\frac{\tan \theta \cos \theta}{\csc \theta}=1 $$

Which expressions are equivalent? 1\. \(\cos \theta\) \(\quad\) II. \(\cos (-\theta)\) \(\quad\) III. \(\frac{\sin (-\theta)}{\tan (-\theta)}\) A. I and II only \(\quad\) B. II and III only \(\quad\) C. I and III only \(\quad\) D. \(I,III\) and III

Find the \(x\) -intercepts of the graph of each function. $$ y=2 \cos \theta+1 $$

In \(\triangle A B C, \angle B\) is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth. $$ a=10, b=15 $$

Verify each identity. $$ \sin ^{2} \theta \tan ^{2} \theta+\cos ^{2} \theta \tan ^{2} \theta=\sec ^{2} \theta-1 $$

If \(\theta\) is in Quadrant \(|\) and \(\tan \theta=\frac{5}{12},\) what is the value of \(\frac{\tan 4 \theta}{5}\) to the nearest hundredth? $$\begin{array}{llll}{\text { A. } 18.10} & {\text { B. } 0.33} & {\text { c. } 0.32} & {\text { D. }-23.90}\end{array}$$

In \(\triangle R S T, m \angle R=37^{\circ}, m \angle T=59^{\circ},\) and \(T S=12\) in. Find \(R S\)

Which expressions are equivalent? \(I .-\tan \left(\frac{\pi}{2}-\theta\right)\) \(\quad\) Il. \(\cos (-\theta)\) \(\quad\) III. \(\tan \left(-\left(\frac{\pi}{2}-\theta\right)\right)\) F. I and II only \(\quad\) G. II and lll only \(\quad\) H. l and \(Il\) only \(\quad\) J. \(I,II,\) and III

Find the measures of the acute angles of a right triangle, to the nearest tenth, if the legs are 135 \(\mathrm{cm}\) and 95 \(\mathrm{cm} .\)

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

If \(\theta\) is in Quadrant \(|\) and \(\sin \theta=\frac{3}{5},\) what is an exact value of \(\sin 2 \theta ?\) $$ \begin{array}{llll}{\text { F. } \frac{9}{25}} & {\text { G. } \frac{24}{25}} & {\text { H. } \frac{6}{5}} & {\text { J. } 73.7}\end{array} $$

In \(\triangle J K L, m \angle L=71^{\circ}, j=11 \mathrm{m},\) and \(m \angle K=46^{\circ} .\) Find \(k\)

Use a half-angle identity to find an exact value of \(\sin 67.5^{\circ} .\)

Which expression is equal to \(\cos 50^{\circ} ?\) A. \(\sin 20^{\circ} \cos 30^{\circ}+\cos 20^{\circ} \sin 30^{\circ} \quad\) B. \(\sin 20^{\circ} \cos 30^{\circ}-\cos 20^{\circ}-\cos 20^{\circ} \sin 30^{\circ}\) \(\mathrm{C} \cdot \cos 20^{\circ} \cos 30^{\circ}+\sin 20^{\circ} \sin 30^{\circ} \quad\) D. \(\cos 20^{\circ} \cos 30^{\circ}-\sin 20^{\circ} \sin 30^{\circ}\)

Find the complete solution of each equation. Express your answer in degrees. \(\sec ^{2} \theta+\sec \theta=0\)

Find the \(x\) -intercepts of the graph of each function. $$ y=\cos ^{2} \theta-1 $$

In \(\triangle A B C, \angle B\) is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth. $$ a=4.1, b=9.4 $$

Simplify each trigonometric expression. $$ (1-\sin \theta)(1+\sin \theta) \csc ^{2} \theta+1 $$

Which expression is NOT equivalent to \(\cos \theta ?\) \(\begin{array}{llll}{\text { F. }-\sin \left(\theta-90^{\circ}\right)} & {\text { G. }-\cos (-\theta)} & {\text { H. } \sin \left(\theta+90^{\circ}\right)} & {\text { J. }-\cos \left(\theta+180^{\circ}\right)}\end{array}\)

Physics When a ray of light passes from one medium into a second, the angle of incidence \(\theta_{1}\) and the angle of refraction \(\theta_{2}\) are related by Snell's law: \(n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2},\) where \(n_{1}\) is the index of refraction of the first medium and \(n_{2}\) is the index of refraction of the second medium. How are \(\theta_{1}\) and \(\theta_{2}\) related if \(n_{2}>n_{1} ?\) If \(n_{2}

In \(\triangle D E F, m \angle F=91^{\circ}, d=17 \mathrm{mm},\) and \(f=21 \mathrm{mm} .\) Find \(m \angle D\)

Find each exact value. Use a sum or difference identity. $$ \cos 405^{\circ} $$

Which expression is an exact value for \(\sin 15^{\circ} ?\) \(\begin{array}{ll}{\text { A. } \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2} \cdot \frac{1}{2}} & {\text { B. } \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2} \cdot \frac{1}{2}}\end{array}\) C. \(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}+\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} \quad\) D. \(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}-\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\)

Find the complete solution of each equation. Express your answer in degrees. \(\cot \theta=\cot ^{2} \theta\)

Sketch one cycle of the graph of each sine function. $$ y=4 \sin \theta $$

Which expression is NOT equal to the other three expressions? \(\begin{array}{llll}{\text { A. } \frac{2}{\tan \theta}} & {\text { B. } \frac{\cot \theta}{\frac{1}{2}}} & {\text { C. } \frac{\sin \theta}{\frac{1}{2} \cos \theta}} & {\text { D. } \frac{2 \cos \theta}{\sin \theta}}\end{array}\)

Identify the period and tell where two asymptotes occur for each function. $$ y=\tan 0.5 \theta $$

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

Find an exact value for sin \(165^{\circ} .\) Show your work.

Find the complete solution of each equation. Express your answer in degrees. \(\sin ^{2} \theta+5 \sin \theta=0\)

Find the \(x\) -intercepts of the graph of each function. $$ y=2 \cos ^{2} \theta-3 \cos \theta-2 $$

Sketch one cycle of the graph of each sine function. $$ y=4 \sin \pi \theta $$

Which equation is NOT true? \(\begin{array}{ll}{\mathbf{F} \cos ^{2} \theta=1-\sin ^{2} \theta} & {\text { G. } \cot ^{2} \theta=\csc ^{2} \theta-1} \\ {\text { H. } \sin ^{2} \theta=\cos ^{2} \theta-1} & {\text { I. } \tan ^{2} \theta=\sec ^{2} \theta-1}\end{array}\)

Identify the period and tell where two asymptotes occur for each function. $$ y=\tan \frac{3 \pi \theta}{2} $$

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

Use the fact that \(\frac{\pi}{6}=\frac{\pi}{2}-\frac{\pi}{3}\) to find an exact value for \(\cos \frac{\pi}{6} .\) Show your work.

Graph each function in the interval from 0 to 2\(\pi\). \(y=\sin (x-\pi)+4\)

Writing Describe the similarities and differences in solving the equations \(4 x+1=3\) and \(4 \sin \theta+1=3\)

Sketch one cycle of the graph of each sine function. $$ y=\sin 4 \theta $$

Which expressions are equivalent? 1\. \((\sin \theta)(\csc \theta-\sin \theta)\) II. \(\sin ^{2} \theta-1\) III. \(\cos ^{2} \theta\) A. I and II only B. II and ll only C.l and III only D. \(1,11,\) and \(\| 1\)