Chapter 7

Because the trigonometric functions are periodic, if a basic trigonometric equation has one solution, it has __________ (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: \(\sin 2 x=\) ______.

We can use identities to help us solve trigonometric equations. Using a Pythagorean identity we see that the equation \(\sin x+\sin ^{2} x+\cos ^{2} x=1\) is equivalent to the basic equation _____ whose solutions are \(x=\) _____.

An equation is called an identity if it is valid for __________ values of the variable. The equation \(2 x=x+x\) is an algebraic identity, and the equation \(\sin ^{2} x+\cos ^{2} x=\) ____________ is a trigonometric identity.

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.

If we know the value of \(\cos x\) and the quadrant in which \(x / 2\) lies, we can find the value of \(\sin (x / 2)\) by using the _______. Formula for sine. State the formula: \(\sin (x / 2)=\) ____________.

If we know the values of the sine and cosine of \(x\) and \(y,\) we can find the value of \(\cos (x-y)\) by using the _____ Formula for cosine. State the formula: \(\cos (x-y)=\) _________.

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

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

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. $$\sin 75^{\circ}$$

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

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

Solve the given equation. $$\sin ^{2} \theta=4-2 \cos ^{2} \theta$$

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

Solve the given equation. $$\sin \theta=\frac{\sqrt{3}}{2}$$

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

Solve the given equation. $$\tan ^{2} \theta-2 \sec \theta=2$$

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

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

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

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

Solve the given equation. $$\csc ^{2} \theta=\cot \theta+3$$

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$\cos 195^{\circ}$$

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

Solve the given equation. $$\cos \theta=-1$$

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

Solve the given equation. $$2 \sin 2 \theta-3 \sin \theta=0$$

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. $$\tan ^{2} x-\sec ^{2} x$$

Solve the given equation. $$\cos \theta=\frac{\sqrt{3}}{2}$$

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\sec x=2, \quad x\) in Quadrant IV

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. $$\tan 165^{\circ}$$

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

Solve the given equation. $$\cos \theta=\frac{1}{4}$$

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

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. $$\sin \frac{19 \pi}{12}$$

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

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

Solve the given equation. $$\cos 2 \theta=\cos ^{2} \theta-\frac{1}{2}$$

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

Solve the given equation. $$\sin \theta=-0.45$$

Solve the given equation. $$2 \sin ^{2} \theta-\cos \theta=1$$

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

Solve the given equation. $$\cos \theta=0.32$$

Solve the given equation. $$\tan \theta-3 \cot \theta=0$$

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

Solve the given equation. $$\sin \theta-1=\cos \theta$$