# 13 Double-Angle, Half-Angle, and Reduction Formulas

### Learning Objectives

In this section, you will:

- Use double-angle formulas to find exact values.
- Use double-angle formulas to verify identities.
- Use reduction formulas to simplify an expression.
- Use half-angle formulas to find exact values.

Bicycle ramps made for competition (see (Figure 1)) must vary in height depending on the skill level of the competitors. For advanced competitors, the angle formed by the ramp and the ground should be such that The angle is divided in half for novices. What is the steepness of the ramp for novices? In this section, we will investigate three additional categories of identities that we can use to answer questions such as this one.

### Using Double-Angle Formulas to Find Exact Values

In the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another look at those same formulas. The double-angle formulas are a special case of the sum formulas, where Deriving the double-angle formula for sine begins with the sum formula,

If we let then we have

Deriving the double-angle for cosine gives us three options. First, starting from the sum formula, and letting we have

Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more variations. The first variation is:

The second variation is:

Similarly, to derive the double-angle formula for tangent, replacing in the sum formula gives

### Double-Angle Formulas

The double-angle formulas are summarized as follows:

### How To

**Given the tangent of an angle and the quadrant in which it is located, use the double-angle formulas to find the exact value.
**

- Draw a triangle to reflect the given information.
- Determine the correct double-angle formula.
- Substitute values into the formula based on the triangle.
- Simplify.

### Using a Double-Angle Formula to Find the Exact Value Involving Tangent

Given that and is in quadrant II, find the following:

## Show Solution

If we draw a triangle to reflect the information given, we can find the values needed to solve the problems on the image. We are given such that is in quadrant II. The tangent of an angle is equal to the opposite side over the adjacent side, and because is in the second quadrant, the adjacent side is on the *x*-axis and is negative. Use the Pythagorean Theorem to find the length of the hypotenuse:

Now we can draw a triangle similar to the one shown in (Figure 2).

- Letâ€™s begin by writing the double-angle formula for sine.
We see that we to need to find and Based on (Figure 2), we see that the hypotenuse equals 5, so and Substitute these values into the equation, and simplify.

Thus,

- Write the double-angle formula for cosine.
Again, substitute the values of the sine and cosine into the equation, and simplify.

- Write the double-angle formula for tangent.
In this formula, we need the tangent, which we were given as Substitute this value into the equation, and simplify.

### Try It

Given with in quadrant I, find

## Show Solution

### Using the Double-Angle Formula for Cosine without Exact Values

Use the double-angle formula for cosine to writein terms of

## Show Solution

#### Analysis

This example illustrates that we can use the double-angle formula without having exact values. It emphasizes that the pattern is what we need to remember and that identities are true for all values in the domain of the trigonometric function.