# 3.3.7: Finding Exact Trigonometric Values Using Sum and Difference Formulas

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Convert angles to sum or difference of 30, 45, and 60 degrees to solve.

You measure an angle with your protractor to be $$165^{\circ}$$. How could you find the exact sine of this angle without Using a calculator?

### Sum and Difference Formulas

You know that $$\sin 30^{\circ} =\dfrac{1}{2}$$, $$\cos 135^{\circ} =−\dfrac{\sqrt{2} }{2}$$, $$\tan 300^{\circ} =−\sqrt{3}$$, etc... from the special right triangles. In this concept, we will learn how to find the exact values of the trig functions for angles other than these multiples of 30^{\circ} ,45^{\circ} , and 60^{\circ} . Using the Sum and Difference Formulas, we can find these exact trig values.

#### Sum and Difference Formulas

\begin{aligned} \sin (a\pm b)&=\sin a\cos b\pm \cos a\sin b \\ \cos (a\pm b)&=\cos a\cos b\mp \sin a\sin b \\ \tan (a\pm b)&=\dfrac{\tan a\pm \tan b}{1\mp \tan a\tan b}\end{aligned}

Let's find the following exact values Using the Sum and Difference Formulas.

1. $$\sin 75^{\circ}$$

This is an example of where we can use the sine sum formula from above, $$\sin (a+b)=\sin a\cos b+\cos a\sin b$$, where $$a=45^{\circ}$$ and $$b=30^{\circ}$$.

\begin{aligned} \sin 75^{\circ} &=\sin (45^{\circ} +30^{\circ} ) \\&=\sin 45^{\circ} \cos 30^{\circ} +\cos 45^{\circ} \sin 30^{\circ} \\ &=\dfrac{\sqrt{2} }{2}\cdot \dfrac{\sqrt{3} }{2}+\dfrac{\sqrt{2} }{2}\cdot \dfrac{1}{2} \\ &=\dfrac{\sqrt{6}+\sqrt{2}}{4} \end{aligned}

In general, $$\sin (a+b)\neq \sin a+\sin b$$ and similar statements can be made for the other sum and difference formulas.

1. $$\cos \dfrac{11 \pi}{12}$$

For this problem, we could use either the sum or difference cosine formula, $$\dfrac{11 \pi}{12}=\dfrac{2\pi}{3}+\dfrac{\pi}{4}$$ or $$\dfrac{11 \pi}{12}=\dfrac{7\pi}{6}−\dfrac{\pi}{4}$$. Let’s use the sum formula.

\begin{aligned} \cos \dfrac{11 \pi}{12}&=\cos (\dfrac{2\pi}{3}+\dfrac{\pi}{4}) \\ &=\cos \dfrac{2\pi}{3}\cos \dfrac{\pi}{4}−\sin \dfrac{2\pi}{3}\sin \dfrac{\pi}{4} \\&=−\dfrac{1}{2}\cdot \dfrac{\sqrt{2} }{2}−\dfrac{\sqrt{3} }{2}\cdot \dfrac{\sqrt{2} }{2} \\&=−\dfrac{\sqrt{2} +\sqrt{6}}{4} \end{aligned}

1. $$\tan \left(−\dfrac{\pi }{12}\right)$$

This angle is the difference between $$\dfrac{\pi}{4}$$ and $$\dfrac{\pi}{3}$$.

\begin{aligned} \tan (\dfrac{\pi}{4}−\dfrac{\pi}{3})&=\dfrac{\tan \dfrac{\pi}{4}−\tan \dfrac{\pi}{3}}{1+\tan \dfrac{\pi}{4}\tan \dfrac{\pi}{3}} \\ &=\dfrac{1−\sqrt{3}}{1+\sqrt{3}}\end{aligned}

This angle is also the same as $$\dfrac{23\pi}{12}$$. You could have also used this value and done $$\tan \left(\dfrac{\pi}{4}+\dfrac{5 \pi}{3}\right)$$ and arrived at the same answer.

##### Example $$\PageIndex{1}$$

Earlier, you were asked to find the exact value of $$\sin 165^{\circ}$$ without Using the calculator.

Solution

We can use the sine sum formula, $$\sin (a+b)=\sin a\cos b+\cos a\sin b$$, where $$a=120^{\circ}$$ and $$b=45^{\circ}$$.

\begin{aligned} \sin 165^{\circ} &=\sin \left(120^{\circ}+45^{\circ}\right) \\ &=\sin 120^{\circ} \cos 45^{\circ}+\cos 120^{\circ} \sin 45^{\circ} \\ &=\dfrac{\sqrt{3}}{2} \cdot \dfrac{\sqrt{2}}{2}+\dfrac{-1}{2} \cdot \dfrac{\sqrt{2}}{2} \\ &=\dfrac{\sqrt{6}-\sqrt{2}}{4} \end{aligned}

##### Example $$\PageIndex{2}$$

Find the exact value of $$\cos 15^{\circ}$$.

Solution

\begin{aligned} \cos 15^{\circ} &=\cos \left(45^{\circ}-30^{\circ}\right) \\ &=\cos 45^{\circ} \cos 30^{\circ}+\sin 45^{\circ} \sin 30^{\circ} \\ &=\dfrac{\sqrt{2}}{2} \cdot \dfrac{\sqrt{3}}{2}+\dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2} \\ &=\dfrac{\sqrt{6}+\sqrt{2}}{4} \end{aligned}

##### Example $$\PageIndex{3}$$

Find the exact value of $$\tan 255^{\circ}$$.

Solution

\begin{aligned} \tan \left(210^{\circ}+45^{\circ}\right) &=\dfrac{\tan 210^{\circ}+\tan 45^{\circ}}{1-\tan 210^{\circ} \tan 45^{\circ}} \\ &=\dfrac{\dfrac{\sqrt{3}}{3}+1}{1-\dfrac{\sqrt{3}}{3}}=\dfrac{\dfrac{\sqrt{3}+3}{3}}{\dfrac{3-\sqrt{3}}{3}}=\dfrac{\sqrt{3}+3}{3-\sqrt{3}} \end{aligned}

## Review

Find the exact value of the following trig functions.

1. $$\sin 15^{\circ}$$
2. $$\cos \dfrac{5 \pi}{12}$$
3. $$\tan 345^{\circ}$$
4. $$\cos (−255^{\circ} )$$
5. $$\sin \dfrac{13 \pi}{12}$$
6. $$\sin \dfrac{17\pi}{12}$$
7. $$\cos 15^{\circ}$$
8. $$\tan (−15^{\circ} )$$
9. $$\sin 345^{\circ}$$
10. Now, use $$\sin 15^{\circ}$$ from #1, and find $$\sin 345^{\circ}$$. Do you arrive at the same answer? Why or why not?
11. Using $$\cos 15^{\circ}$$ from #7, find $$\cos 165^{\circ}$$. What is another way you could find $$\cos 165^{\circ}$$?
12. Describe any patterns you see between the sine, cosine, and tangent of these “new” angles.
13. Using your calculator, find the $$\sin 142^{\circ}$$. Now, use the sum formula and your calculator to find the $$\sin 142^{\circ}$$ Using $$83^{\circ}$$ and $$59^{\circ}$$.
14. Use the sine difference formula to find $$\sin 142^{\circ}$$ with any two angles you choose. Do you arrive at the same answer? Why or why not?
15. Challenge Using $$\sin (a+b)=\sin a\cos b+\cos a\sin b$$ and $$\cos (a+b)=\cos a\cos b−\sin a\sin b$$, show that $$\tan (a+b)=\dfrac{\tan a+\tan b}{1−\tan a\tan b}$$.