Chapter 14

Geometry The lengths of the adjacent sides of a parallelogram are 54 \(\mathrm{cm}\) and 78 \(\mathrm{cm}\) . The larger angle measures \(110^{\circ} .\) What is the length of the longer diagonal? Round your answer to the nearest centimeter.

Find the complete solution in radians of each equation. $$ \sin ^{2} \theta-1=\cos ^{2} \theta $$

Show that cos \(A\) defined as a ratio equals cos \(\theta\) using the unit circle.

Rewrite each expression as a trigonometric function of a single angle measure. $$ \cos 2 \theta \cos 3 \theta-\sin 2 \theta \sin 3 \theta $$

For which set of given information can you compute the area of a triangle? F. Given: the length of one side and the measure of the angle opposite it G. Given: the length of one side and the measure of an angle adjacent to it H. Given: the lengths of two sides and the measure of a nonincluded angle J. Given: the lengths of two sides and the measure of the included angle

Verify each identity. $$ \sin \theta \cos \theta(\tan \theta+\cot \theta)=1 $$

Simplify each expression. $$ \frac{\cos 2 \theta}{\sin \theta+\cos \theta} $$

Geometry The lengths of the adjacent sides of a parallelogram are 21 \(\mathrm{cm}\) and 14 \(\mathrm{cm} .\) The smaller angle measures \(58^{\circ} .\) What is the length of the shorter diagonal? Round your answer to the nearest centimeter.

The bases on a baseball diamond form a square 90 ft on a side. The pitcher's plate is 60 \(\mathrm{ft}\) 6 in. from the back corner of home plate. a. About how far is the pitcher's plate from second base? b. A line drive is 10 ft high when it passes over the third baseman, who is 100 ft from home plate. At what angle did the ball leave the bat? (Assume the ball is 4 \(\mathrm{ft}\) above the ground when it is hit.)

Find the complete solution in radians of each equation. $$ 2 \sin \theta+1=\csc \theta $$

Rewrite each expression as a trigonometric function of a single angle measure. $$ \frac{\tan 5 \theta+\tan 6 \theta}{1-\tan 5 \theta \tan 6 \theta} $$

A surveyor picks two points 250 \(\mathrm{m}\) apart in front of a tall building. The angle of elevation from one point is \(37^{\circ} .\) The angle of elevation from the other point is \(13^{\circ} .\) What is the best estimate for the height of the building? \(\begin{array}{lllll}{\text { A. } 150 \mathrm{m}} & {\text { B. } 138 \mathrm{m}} & {\text { C. } 83 \mathrm{m}} & {\text { D. } 56 \mathrm{m}}\end{array}\)

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

Open-Ended Choose an angle measure \(A .\) a. Find \(\sin A\) and \(\cos A .\) b. Use an identity to find \(\sin 2 A\) c. Use an identity to find \(\cos \frac{A}{2}\)

Critical thinking Does the Law of Cosines apply to a right triangle? That is does \(c^{2}=a^{2}+b^{2}-2 a b \cos C\) remain true when \(\angle C\) is a right angle? Justify your answer.

Use the definitions of trigonometric ratios in right \(\triangle A B C\) to verify each identity. \(\sec A=\frac{1}{\cos A}\)

Find the complete solution in radians of each equation. $$ 3 \tan ^{2} \theta-1=\sec ^{2} \theta $$

Rewrite each expression as a trigonometric function of a single angle measure. $$ \frac{\tan 3 \theta-\tan \theta}{1+\tan 3 \theta \tan \theta} $$

Two sides of a scalene triangle are 9 \(\mathrm{m}\) and 14 \(\mathrm{m}\) . The area of the triangle is 31.5 \(\mathrm{m}^{2} .\) Find the measure of one of the angles of the triangle to the nearest tenth of a degree. Show your work.

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

Writing Is \(\frac{\tan \theta}{4}=\tan \frac{\theta}{4}\) an identity? Explain.

Physics A pendulum 36 in. long swings \(30^{\circ}\) from the vertical. How high above the lowest position is the pendulum at the end of its swing? Round your answer to the nearest tenth of an inch.

Use the definitions of trigonometric ratios in right \(\triangle A B C\) to verify each identity. \(\tan A=\frac{\sin A}{\cos A}\)

Find the complete solution in radians of each equation. $$ \sin \theta \cos \theta=\frac{1}{2} \cos \theta $$

a. Graph \(y=\sin 2 x\) and \(y=2 \sin x\) on the same axes. b. Does sin \(2 x=2 \sin x\) for all values of \(x ?\) Is sin \(2 x=2 \sin x\) an identity? Explain. c. Does \(\sin 2 x=2 \sin x\) for any values of \(x ?\) If so, what are they? d. Open-Ended Find an equation of the form \(a \sin b=c \sin d\) whose solutions are 2\(\pi n .\)

Verify each identity. $$ (\cot \theta+1)^{2}=\csc ^{2} \theta+2 \cot \theta $$

Use double-angle identities to write each expression, using trigonometric functions of \(\theta\) instead of 4\(\theta .\) $$ \sin 4 \theta $$

Use the definitions of trigonometric ratios in right \(\triangle A B C\) to verify each identity. \(\cos ^{2} A+\sin ^{2} A=1\)

Find the complete solution in radians of each equation. $$ \tan \theta \sin \theta=3 \sin \theta $$

Gears The diagram at the right shows a gear whose radius is 10 \(\mathrm{cm} .\) Point \(A\) represents a \(60^{\circ}\) counterclockwise rotation of point \(P(10,0)\) . Point \(B\) represents a \(\theta\) -degree rotation of point \(A\) . The coordinates of \(B\) are \(\left(10 \cos \left(\theta+60^{\circ}\right), 10 \sin \left(\theta+60^{\circ}\right)\right) .\) Write these coordinates in terms of \(\cos \theta\) and \(\sin \theta.\)

Verify each identity. Express \(\cos \theta \csc \theta \cot \theta\) in terms of \(\sin \theta\)

Use double-angle identities to write each expression, using trigonometric functions of \(\theta\) instead of 4\(\theta .\) $$ \cos 4 \theta $$

a. Critical Thinking A function is even if \(f(-x)=f(x) .\) A function is odd if \(f(-x)=-f(x) .\) Which trigonometric functions are even? Which are odd? b. Writing Are all functions either even or odd? Explain your answer. Give a counterexample if possible.

Find the complete solution in radians of each equation. $$ 2 \cos ^{2} \theta+\sin \theta=1 $$

Find each angle measure to the nearest tenth of a degree. $$ \cos ^{-1} \frac{3}{5} $$

Verify each identity. Express \(\frac{\cos \theta}{\sec \theta+\tan \theta}\) in terms of \(\sin \theta\)

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

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

Find the complete solution in radians of each equation. $$ \sin \theta \cot ^{2} \theta-3 \sin \theta=0 $$

Find each angle measure to the nearest tenth of a degree. $$ \tan ^{-1} 0.4569 $$

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

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

Find the complete solution in radians of each equation. $$ 4 \sin ^{2} \theta+1=4 \sin \theta $$

Find each angle measure to the nearest tenth of a degree. $$ \sin ^{-1} \frac{5}{8} $$

Verify each identity. Writing Describe the similarities and differences in solving an equation and in verifying an identity.

A ladder rests against a vertical building, as shown at the right. The ladder is 14 ft long and forms an angle of \(76.5^{\circ}\) with the ground. Which statement is NOT true? A. The bottom of the ladder is 13.6 \(\mathrm{ft}\) from the base of the building. B. The bottom of the ladder is 3.3 \(\mathrm{ft}\) from the base of the building. C. The top of the ladder is 13.6 \(\mathrm{ft}\) from the ground. D. The ladder forms an angle of \(13.5^{\circ}\) with the building.

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

The sides of a rectangle are 25 \(\mathrm{cm}\) and 8 \(\mathrm{cm}\) . What is the measure of the angle formed by the short side and a diagonal of the rectangle? F. \(17.7^{\circ}\) G. \(18.7^{\circ}\) H. \(71.3^{\circ}\) J. \(72.3^{\circ}\)

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

Find the complete solution in radians of each equation. $$ \tan \theta \cot \theta-\tan \theta+2 \cot \theta=0 $$