10.03 Angle sum and difference identities
Exploration
Consider the statement $\cos90^\circ=\cos90^\circ+\cos0^\circ$cos90°=cos90°+cos0°. Is this equation true?
If we evaluate each of the expressions, we get an equation that says $0=0+1$0=0+1. This statement is not true! What does that tell us about trigonometric functions?
In general, the distributive property does not apply to trigonometric functions. Instead, however, we can apply the sum and difference identities to evaluate sums and differences of angle measures.
$\sin\left(A+B\right)=\sin A\cos B+\cos A\sin B$sin(A+B)=sinAcosB+cosAsinB
$\sin\left(A-B\right)=\sin A\cos B-\cos A\sin B$sin(A−B)=sinAcosB−cosAsinB
$\cos\left(A+B\right)=\cos A\cos B-\sin A\sin B$cos(A+B)=cosAcosB−sinAsinB
$\cos\left(A-B\right)=\cos A\cos B+\sin A\sin B$cos(A−B)=cosAcosB+sinAsinB
$\tan\left(A+B\right)=\frac{\tan A+\tan B}{1-\tan A\tan B}$tan(A+B)=tanA+tanB1−tanAtanB
$\tan\left(A-B\right)=\frac{\tan A-\tan B}{1+\tan A\tan B}$tan(A−B)=tanA−tanB1+tanAtanB
These identities can be proven using the law of sines, which we will visit at a later time. For now, we can accept their truth and use them as shortcuts.
Worked example
Question 1
Evaluate: Find an exact value expression for $\sin15^\circ$sin15°
Think: Notice that $15^\circ=45^\circ-30^\circ$15°=45°−30°. Let's use $A=45^\circ$A=45° and $B=30^\circ$B=30° in the difference formula for the $\sin\left(A-B\right)$sin(A−B).
Do: Substitute into the formula and evaluate the expressions.
| $\sin15^\circ$sin15° | $=$= | $\sin\left(45^\circ-30^\circ\right)$sin(45°−30°) |
Substitute$15^\circ=45^\circ-30^\circ$15°=45°−30° . |
| $=$= | $\sin45^\circ\cos30^\circ-\cos45^\circ\sin30^\circ$sin45°cos30°−cos45°sin30° |
Let $A=45^\circ$A=45° and $B=30^\circ$B=30° in the difference formula. |
|
| $=$= | $\frac{1}{\sqrt{2}}\frac{\sqrt{3}}{2}-\frac{1}{2}\times\frac{1}{\sqrt{2}}$1√2√32−12×1√2 |
Substitute the known exact values. |
|
| $=$= | $\frac{\sqrt{3}-1}{2\sqrt{2}}$√3−12√2 |
Simplify. |
|
| $=$= | $\frac{\sqrt{6}-\sqrt{2}}{4}$√6−√24 |
Rationalize the denominator. |
Practice questions
Question 2
Given that $\sin A=\frac{24}{25}$sinA=2425 and $\cos B=\frac{20}{29}$cosB=2029, find the exact value of:
$\sin\left(A+B\right)$sin(A+B)
$\cos\left(A-B\right)$cos(A−B)
Question 3
Using the expansion of $\cos\left(A+B\right)$cos(A+B), find the exact value of $\cos75^\circ$cos75°. Express the value in rationalized form.
Question 4
Find the exact value of $\tan\left(\frac{\pi}{12}\right)$tan(π12) in simplest radical form. Express the value in rationalized form.