# Trigonometry : Identities of Doubled Angles

## Example Questions

### Example Question #1 : Identities Of Doubled Angles

Simplify the function below:

Explanation:

We need to use the following formulas:

a)

b)

c)

We can simplify  as follows:

### Example Question #2 : Identities Of Doubled Angles

Given , what is  in terms of ?

Explanation:

To solve this problem, we need to use the formula:

Substituting , we get

### Example Question #3 : Identities Of Doubled Angles

Using trigonometric identities, determine whether the following is valid:

Only in the range of:

Uncertain

False

Only in the range of:

True

True

Explanation:

In order to prove this trigonometric equation we can work with either the left or right side of the equation and attempt to make them equal. We will choose to work with the left side of the equation. First we separate the fractional term:

We separated the fractional term because we notice we have a double angle. Recalling our trigonometric identities, the fractional term is the inverse of the power reducing formula for sine.

Now separating out the sine terms:

Now recalling the basic identities:

Using the trigonometric identities we have proven that the equation is true.

### Example Question #4 : Identities Of Doubled Angles

Using trigonometric identities determine whether the following is true:

False

Only in the range of:

Only in the range of:

Uncertain

True

True

Explanation:

We choose which side to work with in the given equation. Selecting the right hand side since it contains a double angle we attempt to use the double angle formula to determine the equivalence:

Next we reduce and split the fraction as follows:

Recalling the basic identities:

This proves the equivalence.

### Example Question #4 : Identities Of Doubled Angles

Using a double angle formula, find the value of

.

Explanation:

The formula for a doubled angle with sine is

Plug in our given value and solve.

Combine our terms.

### Example Question #5 : Identities Of Doubled Angles

Find the value of  if , and if the value of  is less than zero.

Explanation:

Write the Pythagorean Identity.

Substitute the value of  and solve for .

Since the  must be less than zero, choose the negative sign.

Write the double-angle identity of .

Substitute the known values.

### Example Question #7 : Identities Of Doubled Angles

Suppose  is an angle in the third quadrant, such that:

What is the value of ?

Explanation:

We can exploit the following trigonometric identity:

Then we can do:

With this value we can conveniently find our solution to be:

### Example Question #6 : Identities Of Doubled Angles

What is the period of ?

Explanation:

The key here is to double-angle identity for to simplify the function.

In this case, , which means...

From there, we can use the fact that the period of or is . Consequently,

### Example Question #9 : Identities Of Doubled Angles

Expand the following expression using double-angle identities.

Explanation:

Since

and ,

then .

Here we have to use the double-angle identities for both sine and cosine,

and .

Using these identities:

Using the distributive property: