The sum and difference identities of angles are trigonometric identities used to calculate the values of certain angles. These identities can be used to rewrite the angles as a sum or subtraction of common angles. For example, to calculate the sine or cosine of 15°, we can rewrite 15 as the subtraction of 45 and 30 since the values of the sines and cosines of 45° and 30° are the most common and are generally known.
Here, we will learn to derive the sum and difference identities of angles and apply them to solve some practice problems.
TRIGONOMETRY
Relevant for…
Learning about the sum and difference identities with examples.
TRIGONOMETRY
Relevant for…
Learning about the sum and difference identities with examples.
What are the sum and difference identities?
The sum and difference identities of angles are trigonometric identities, which can be used to find the values of trigonometric functions of any angle.
However, the most practical use of these identities is to find the exact values of an angle that can be written as a sum or difference of familiar values for sine, cosine, and tangent of the angles 30°, 45°, 60°, 90°, and its multiples.
The identities of the sum of angles are:
| $latex \sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$ $latex \cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$ $latex \tan(A+B)=\frac{\tan(A)+\tan(B)}{1-\tan(A)\tan(B)}$ |
The identities of the difference of angles are:
| $latex \sin(A-B)=\sin(A)\cos(B)-\cos(A)\sin(B)$ $latex \cos(A-B)=\cos(A)\cos(B)+\sin(A)\sin(B)$ $latex \tan(A-B)=\frac{\tan(A)-\tan(B)}{1+\tan(A)\tan(B)}$ |
Half-angle identities – Examples with answers
The following examples are solved by applying the sum and difference identities of sine, cosine, and tangent angles. Try to solve the problems yourself before looking at the answer.
EXAMPLE 1
Find the exact value of the cosine of 75° using an angle sum identity.
Solution
We can use the identity of the sum of cosine angles since 75° is equal to the sum of 30° and 45°. Therefore, we have:
$$\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$$
$$\cos(30+45)=\cos(30)\cos(45)-\sin(30)\sin(45)$$
$latex =\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}-\frac{1}{2}\cdot \frac{\sqrt{2}}{2}$
$latex =\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}$
$latex =\frac{\sqrt{6}-\sqrt{2}}{4}$
The value of the cosine of 75° is $latex \frac{\sqrt{6}-\sqrt{2}}{4}$.
EXAMPLE 2
Use an angle difference identity to find the exact value of the sine of 15°.
Solution
We can use the identity of the difference of angles for sines since 15° is equal to the difference of 45° and 30°. Therefore, we have:
$$\sin(A-B)=\sin(A)\cos(B)-\cos(A)\sin(B)$$
$$\sin(45-30)=\sin(45)\cos(30)-\cos(45)\sin(30)$$
$latex =\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\cdot \frac{1}{2}$
$latex =\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}$
$latex =\frac{\sqrt{6}-\sqrt{2}}{4}$
This means that the exact value of the sine of 15° is $latex \frac{\sqrt{6}-\sqrt{2}}{4}$.
EXAMPLE 3
Find the exact value of $latex \cos (\frac{7 \pi}{12})$ using an angle sum identity.
Solution
We use the sum of cosines identity since we can express the angle as a sum of known angles:
$latex \cos(\frac{7\pi}{12})=\cos(\frac{4\pi}{12}+\frac{3\pi}{12})$
$latex =\cos(\frac{\pi}{3}+\frac{\pi}{4})$
Using these angles in the formula, we have:
$$\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$$
$$\cos(\frac{\pi}{3}+\frac{\pi}{4})=\cos(\frac{\pi}{3})\cos(\frac{\pi}{4})-\sin(\frac{\pi}{3})\sin(\frac{\pi}{4})$$
$latex =\frac{1}{2} \cdot \frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2}$
$latex =\frac{\sqrt{2}}{4}-\frac{\sqrt{6}}{4}$
$latex =\frac{\sqrt{2}-\sqrt{6}}{4}$
This means that the exact value of the cosine of $latex \frac{7\pi}{12}$ is $latex \frac{\sqrt{2}-\sqrt{6}}{4}$.
Sum and difference identities – Practice problems
Apply the formulas for the sum and difference identities of sine and cosine angles to solve the following practice problems. Select an answer and check it to see if you got the correct answer.
See also
Interested in learning more about trigonometric identities? Take a look at these pages:
Jefferson Huera Guzman
Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.
