# Algebra II: Trigonometric Identities

On this page we hope to clear up problems that you might have with the trigonometric identities, such as the double angle identities and the half-angle identities.  Read on or follow any of the links below to better your understanding of the trig. identities.

Cosines
Other identities
Double angle identities
Proving identities
Quiz on trigonometric identities

## Cosines

The trig. identities are important identities that involve sums or differences of angles.  There is an identity that shows that the cosine of the difference of two angles is related to the cosines and sines of the angles themselves.  This identity is given below (A and B are used in place of alpha and beta, respectively since text only browsers do not support Greek characters).

cos (A - B) = (cos A)(cos B) = (sin A)(sin B)

The accompanying figure gives a graphical representation of the cosine identity.

This identity is useful when you are asked to find the cosine of a non-30°-45°-60°-90° angle, such as 10°.

Example:

```1.  Problem: Find the cos 15°.
Solution: Write 15° in terms of angles with known trig.
ratio values.

cos (45° - 30°)

Use the cosine identity to rewrite the expression.

(cos 45°)cos 30° + (sin 45°)sin 30°

Using the values you know for the trig. ratios of special angles,
rewrite the expression.

SQRT(2)   SQRT(3)   SQRT(2)   1
------- * ------- + ------- * -
2         2         2      2

Perform the indicated multiplications.

SQRT(6)   SQRT(2)
------- + -------
4         4

SQRT(6) + SQRT(2)
-----------------
4
```
There is also a cosine identity for a sum of angles.  It is shown below.

cos (A + B) = (cos A)cos B - (sin A)sin B

## The Other Identities

There are also sine identities and tangent identities.  They are listed below.

sin (A + B) = (sin A)(cos B) + (cos A)(sin B)

sin (A - B) = (sin A)(cos B) - (cos A)(sin B)

tan (A + B) = (tan A + tan B)/(1 - (tan A)(tan B))

tan (A - B) = (tan A - tan B)/(1 + (tan A)(tan B))

Example:

```1.  Problem: Find tan 15°.
Solution: Rewrite as a difference of angles with known trig. ratio values.

tan (45° - 30°)

Use the tangent of differences identity to rewrite the
expression.

tan 45° - tan 30°
--------------------
1 + (tan 45°)(tan 30°)

Substitute the known trig. ratio values in and perform the
indicated operations.

1 - (SQRT(3))/3
---------------
1 + (SQRT(3))/3

3 - SQRT(3)
-----------
3 + SQRT(3)

2 - SQRT(3)
```

## Double Angle Identities

Identities involving sin 2x or cos 2x are called double-angle identities.  These identities are derived using the sum and difference identities.  Below, we will show you how one of the double-angle identities was derived.
```            sin 2x = sin (x + x)

Use the sine of sums identity.

(sin x)(cos x) + (cos x)(sin x)

2(sin x)(cos x)
```
Example:
```1.  Problem: If the sine of theta is (3/8) and theta is in the first
quadrant, what is sin 2(theta)?  (Use the
accompanying figure.)
Solution: From the diagram, we see that cos (theta) = (SQRT(55)/8.

Use the double angle identity for sine.

sin 2(theta) = 2(sin (theta))(cos (theta))

Plug in the values you know.

2 * (3/8) * (SQRT(55)/8)

Perform the indicated multiplications.

(3(SQRT(55)))/32
```
The other identities are listed below.

cos 2x = cos^2 x - sin^2 x

cos 2x = 1 - 2sin^2 x

cos 2x = 2cos^2 x - 1

tan 2x = (2tan x)/(1 - tan^2 x)

sin^2 x = (1 - cos 2x)/2

cox^2 x = (1 + cos 2x)/2

## Proving Identities

Identities can be used to prove identities equal to other identities.

Example:

```1.  Problem: Prove the following identity:
tan^2 x - sin^2 x = (sin^2 x)(tan^2 x)
Solution: Write each side in terms of sin x and cos x.

sin^2 x                      sin^2 x
------- - sin^2 x = (sin^2 x)-------
cos^2 x                      cos^2 x

Now, only deal with one side of the equation.  Find the common
denominator and subtract.

sin^2 x - (sin^2 x)(cos^2 x)
----------------------------
cos^2 x

Factor out a sin^2 x.

(sin^2 x)(1 - cos^2 x)
----------------------
cos^2 x

Use the Pythagorean Identities
to replace (1 - cos^2 x) with sin^2 x.

(sin^2 x)(sin^2 x)
------------------
cos^2 x

sin^2 x
(sin^2 x)-------
cos^2 x
```

Take the quiz on trigonometric identities.  The quiz is very useful for either review or to see if you've really got the topic down.

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