Chapter 8

If we know the values of the sine and cosine of x and y, we can find the value of \(\sin (x+y)\) by using the __________ Formula for Sine. State the formula: \(\sin (x+y)=\)

Because the trigonometric functions are periodic, if a basic trigonometric equation has one solution, it has (several/infinitely many) solutions._________(several/infinitely many) solutions.

If we know the values of \(\sin x\) and \(\cos x,\) we can find the value of \(\sin 2 x\) by using the _____ Formula for Sine. State the formula: \(\sin 2 x=\) _____.

The basic equation \(\sin x=2\) has__________(no/one/infinitely many) solutions, whereas the basic equation \(\sin x=0.3\) has_________(no/one/infinitely many) solutions.

For any \(x\) it is true that \(\cos (-x)\) has the same value as \(\cos x\). We express this fact as the identity _____.

\(3-16 \cdot\) Solve the given equation. $$ 2 \cos ^{2} \theta+\sin \theta=1 $$

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$ \sin 75^{\circ} $$

\(3-10\) Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. $$ \sin X=\frac{5}{13}, \quad x \text { in Quadrant I } $$

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

\(3-16 \cdot\) Solve the given equation. $$ \sin ^{2} \theta=4-2 \cos ^{2} \theta $$

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$ \sin 15^{\circ} $$

\(3-10\) Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. $$ \tan x=-\frac{4}{3}, \quad x \text { in Quadrant II } $$

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

\(3-16 \cdot\) Solve the given equation. $$ \tan ^{2} \theta-2 \sec \theta=2 $$

\(5-16=\) Solve the given equation. $$ \sin \theta=\frac{\sqrt{3}}{2} $$

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$ \cos 105^{\circ} $$

\(3-10\) 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. $$ \sin \theta \sec \theta $$

\(3-16 \cdot\) Solve the given equation. $$ \csc ^{2} \theta=\cot \theta+3 $$

\(5-16=\) Solve the given equation. $$ \sin \theta=-\frac{\sqrt{2}}{2} $$

\(3-10\) 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. $$ \tan \theta \csc \theta $$

\(3-16 \cdot\) Solve the given equation. $$ 2 \sin 2 \theta-3 \sin \theta=0 $$

\(5-16=\) Solve the given equation. $$ \cos \theta=-1 $$

\(3-10\) Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. $$ \sin x=-\frac{3}{5}, \quad x \text { in Quadrant III } $$

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

\(3-16 \cdot\) Solve the given equation. $$ 3 \sin 2 \theta-2 \sin \theta=0 $$

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$ \tan 165^{\circ} $$

\(3-10\) Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. $$ \sec x=2, \quad x \text { in Quadrant IV } $$

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

\(3-16 \cdot\) Solve the given equation. $$ \cos 2 \theta=3 \sin \theta-1 $$

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$ \sin \frac{19 \pi}{12} $$

\(3-10\) 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. $$ \sin u+\cot u \cos u $$

\(3-16 \cdot\) Solve the given equation. $$ \cos 2 \theta=\cos ^{2} \theta-\frac{1}{2} $$

\(5-16=\) Solve the given equation. $$ \sin \theta=-0.3 $$

\(3-10\) 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. $$ \cos ^{2} \theta\left(1+\tan ^{2} \theta\right) $$

\(3-16 \cdot\) Solve the given equation. $$ 2 \sin ^{2} \theta-\cos \theta=1 $$

\(5-16=\) Solve the given equation. $$ \sin \theta=-0.45 $$

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

\(3-16 \cdot\) Solve the given equation. $$ \tan \theta-3 \cot \theta=0 $$

\(5-16=\) Solve the given equation. $$ \cos \theta=0.32 $$

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$ \sin \left(-\frac{5 \pi}{12}\right) $$

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

\(3-16 \cdot\) Solve the given equation. $$ \sin \theta-1=\cos \theta $$

\(5-16=\) Solve the given equation. $$ \tan \theta=-\sqrt{3} $$

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$ \cos \frac{11 \pi}{12} $$

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

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