Chapter 8

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \cos t \tan t $$

Find all solutions of the equation. $$\cos x+1=0$$

Find the exact value of each expression, if it is defined. (a) \(\sin ^{-1} \frac{1}{2}\) (b) \(\cos ^{-1} \frac{1}{2}\) (c) \(\cos ^{-1} 2\)

\(1-12\) : Use an addition or subtraction formula to find the exact value of the expression, as demonstrated in Example \(1 .\) $$ \sin 75^{\circ} $$

1-8 Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\sin x=\frac{5}{13}, \quad x\) in quadrant I

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \cos t \csc t $$

Find all solutions of the equation. $$\sin x+1=0$$

Find the exact value of each expression, if it is defined. (a) \(\sin ^{-1} \frac{\sqrt{3}}{2}\) (b) \(\cos ^{-1} \frac{\sqrt{3}}{2}\) (c) \(\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)

\(1-12\) : Use an addition or subtraction formula to find the exact value of the expression, as demonstrated in Example \(1 .\) $$ \sin 15^{\circ} $$

1-8 Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\tan x=-\frac{4}{3}, \quad x\) in quadrant \(\Pi\)

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \sin \theta \sec \theta $$

Find all solutions of the equation. $$2 \sin x-1=0$$

Find the exact value of each expression, if it is defined. (a) \(\sin ^{-1} \frac{\sqrt{2}}{2}\) (b) \(\cos ^{-1} \frac{\sqrt{3}}{2}\) (c) \(\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)\)

\(1-12\) : Use an addition or subtraction formula to find the exact value of the expression, as demonstrated in Example \(1 .\) $$ \cos 105^{\circ} $$

1-8 Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\cos x=\frac{4}{5}, \quad \csc x<0\)

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \tan \theta \csc \theta $$

Find all solutions of the equation. $$\sqrt{2} \cos x-1=0$$

\(1-12\) : Use an addition or subtraction formula to find the exact value of the expression, as demonstrated in Example \(1 .\) $$ \cos 195^{\circ} $$

1-8 Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\csc x=4, \quad \tan x<0\)

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \tan ^{2} x-\sec ^{2} x $$

Find all solutions of the equation. $$\sqrt{3} \tan x+1=0$$

Find the exact value of each expression, if it is defined. (a) \(\sin ^{-1} 1\) (b) \(\cos ^{-1} 1\) (c) \(\cos ^{-1}(-1)\)

\(1-12\) : Use an addition or subtraction formula to find the exact value of the expression, as demonstrated in Example \(1 .\) $$ \tan 15^{\circ} $$

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \frac{\sec x}{\csc x} $$

Find all solutions of the equation. $$\cot x+1=0$$

Find the exact value of each expression, if it is defined. (a) \(\tan ^{-1} 1\) (b) \(\tan ^{-1}(-1)\) (c) \(\tan ^{-1} 0\)

1-8 Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\sec x=2, \quad x\) in quadrant \(\mathrm{IV}\)

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \sin u+\cot u \cos u $$

Find all solutions of the equation. $$4 \cos ^{2} x-1=0$$

Find the exact value of each expression, if it is defined. (a) \(\tan ^{-1} \frac{\sqrt{3}}{3}\) (b) \(\tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right)\) (c) \(\sin ^{-1}(-2)\)

1-8 Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\tan x=-\frac{1}{3}, \quad \cos x>0\)

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \cos ^{2} \theta\left(1+\tan ^{2} \theta\right) $$

Find all solutions of the equation. $$2 \cos ^{2} x-1=0$$

Find the exact value of each expression, if it is defined. (a) \(\sin ^{-1} 0\) (b) \(\cos ^{-1} 0\) (c) \(\cos ^{-1}\left(-\frac{1}{2}\right)\)

1-8 Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\cot x=\frac{2}{3}, \quad \sin x>0\)

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \frac{\sec \theta-\cos \theta}{\sin \theta} $$

Find all solutions of the equation. $$\sec ^{2} x-2=0$$

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. (a) \(\sin ^{-1}(0.13844)\) (b) \(\cos ^{-1}(-0.92761)\)

9–14 Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine, as in Example 4. $$\sin ^{4} x$$

Write the trigonometric expression in terms of sine and cosine, and then simplify. $$ \frac{\cot \theta}{\csc \theta-\sin \theta} $$

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. (a) \(\cos ^{-1}(0.31187)\) (b) \(\tan ^{-1}(26.23110)\)

Find all solutions of the equation. $$\csc ^{2} x-4=0$$

Simplify the trigonometric expression. $$ \frac{\sin x \sec x}{\tan x} $$

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. (a) \(\tan ^{-1}(1.23456)\) (b) \(\sin ^{-1}(1.23456)\)

Find all solutions of the equation. $$3 \csc ^{2} x-4=0$$

\(1-12\) : Use an addition or subtraction formula to find the exact value of the expression, as demonstrated in Example \(1 .\) $$ \cos \frac{11 \pi}{12} $$

9–14 Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine, as in Example 4. $$\cos ^{2} x \sin ^{4} x$$

Simplify the trigonometric expression. $$ \cos ^{3} x+\sin ^{2} x \cos x $$

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. (a) \(\cos ^{-1}(-0.25713)\) (b) \(\tan ^{-1}(-0.25713)\)

Find all solutions of the equation. $$1-\tan ^{2} x=0$$