# 3.5: Double Angle Identities

## Learning Objectives

- Use the double angle identities to solve other identities.
- Use the double angle identities to solve equations.

## Deriving the Double Angle Identities

One of the formulas for calculating the sum of two angles is:

If and are both the same angle in the above formula, then

This is the double angle formula for the sine function. The same procedure can be used in the sum formula for cosine, start with the sum angle formula:

If and are both the same angle in the above formula, then

This is one of the double angle formulas for the cosine function. Two more formulas can be derived by using the Pythagorean Identity, .

and likewise

Therefore, the double angle formulas for are:

Finally, we can calculate the double angle formula for tangent, using the tangent sum formula:

If and are both the same angle in the above formula, then

## Applying the Double Angle Identities

**Example 1:** If and is in Quadrant II, find , , and .

**Solution:** To use , the value of must be found first.

.

However since is in Quadrant II, is negative or .

For , use

For , use . From above, .

**Example 2:** Find .

**Solution:** Think of as .

Now, use the double angle formulas for both sine and cosine. For cosine, you can pick which formula you would like to use. In general, because we are proving a cosine identity, stay with cosine.

**Example 3:** If and is an acute angle, find the exact value of .

**Solution:** Cotangent and tangent are reciprocal functions, and .

**Example 4:** Given and is in Quadrant I, find the value of .

**Solution:** Using the double angle formula, . Because we do not know , we need to solve for in the Pythagorean Identity, . Substitute this into our formula and solve for .

At this point we need to get rid of the fraction, so multiply both sides by the reciprocal.

Now, this is in the form of a quadratic equation, even though it is a quartic. Set , making the equation . Once we have solved for , then we can substitute back in and solve for . In the Quadratic Formula, .

So, or . This means that or so or .

**Example 5:** Prove

**Solution:** Substitute in the double angle formulas. Use , since it will produce only one term in the numerator.

## Solving Equations with Double Angle Identities

Much like the previous sections, these problems all involve similar steps to solve for the variable. Isolate the trigonometric function, using any of the identities and formulas you have accumulated thus far.

**Example 6:** Find all solutions to the equation in the interval

**Solution:** Apply the double angle formula

The values for in the interval are and and the values for in the interval are and . Thus, there are four solutions.

**Example 7:** Solve the trigonometric equation such that

**Solution:** Using the sine double angle formula:

**Example 8:** Find the exact value of given if is in the second quadrant.

**Solution:** Use the double-angle formula with cosine only.

**Example 9:** Solve the trigonometric equation over the interval .

**Solution:** Pull out a 2 from the left-hand side and this is the formula for .

The solutions for are , dividing each of these by 2, we get the solutions for , which are .

## Points to Consider

- Are there similar formulas that can be derived for other angles?
- Can technology be used to either solve these trigonometric equations or to confirm the solutions?

## Review Questions

- If and is in Quad II, find the exact values of and
- Find the exact value of
- Verify the identity:
- Verify the identity:
- If and is in Quad III, find the exact values of and
- Find all solutions to if
- Find all solutions to if
- If and , use the double angle formulas to determine each of the following:
- Use the double angle formulas to prove that the following equations are identities.
- Solve the trigonometric equation such that
- Solve the trigonometric equation such that
- Prove .
- Solve for in the interval .
- Solve the trigonometric equation such that

## Review Answers

- If and in Quadrant II, then cosine and tangent are negative. Also, by the Pythagorean Theorem, the third side is . So, and . Using this, we can find , and .
- This is one of the forms for .
- Step 1: Use the cosine sum formula Step 2: Use double angle formulas for and Step 3: Distribute and simplify.
- Step 1: Expand using the double angle formula. Step 2: change and find a common denominator.
- If and in Quadrant III, then and (Pythagorean Theorem, ). So,
- Step 1: Expand Step 2: Separate and solve each for .
- Expand and simplify when , and when . Therefore, the solutions are .
- a. b. c.
- a. b. c.
- when and when .
- Note: If we go back to the equation , we can see that must be positive or zero, since is always positive or zero. For this reason, and must have the same sign (or one of them must be zero), which means that cannot be in the second or fourth quadrants. This is why and are not valid solutions.
- Use the double angle identity for .