# Trigonometry : Identities of Halved Angles

## Example Questions

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

Find  if  and .

Explanation:

The double-angle identity for sine is written as

and we know that

Using , we see that , which gives us

Since we know  is between  and , sin  is negative, so . Thus,

.

Finally, substituting into our double-angle identity, we get

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

Find the exact value of  using an appropriate half-angle identity.

Explanation:

The half-angle identity for sine is:

If our half-angle is , then our full angle is . Thus,

The exact value of  is expressed as , so we have

Simplify under the outer radical and we get

Now simplify the denominator and get

Since  is in the first quadrant, we know sin is positive. So,

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

Which of the following best represents ?

Explanation:

Write the half angle identity for cosine.

Replace theta with two theta.

Therefore:

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

What is the amplitude of ?

Explanation:

The key here is to use the half-angle identity for to convert it and make it much easier to work with.

In this case, , so therefore...

Consequently, has an amplitude of .

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

If , then calculate .

Explanation:

Because , we can use the half-angle formula for cosines to determine .

In general,

for .

For this problem,

Hence,

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

What is ?

Explanation:

Let ; then

.

We'll use the half-angle formula to evaluate this expression.

Now we'll substitute  for .

is in the first quadrant, so  is positive. So

.

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

What is , given that  and  are well defined values?

Explanation:

Using the half angle formula for tangent,

,

we plug in 30 for .

We also know from the unit circle that  is  and  is .

Plug all values into the equation, and you will get the correct answer.