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Finding Exact Trigonometric Values Using Sum and Difference Formulas

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Finding Exact Trig Values using Sum and Difference Formulas
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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?

Guidance

You know that $\sin 30^\circ=\frac{1}{2}, \cos 135^\circ=-\frac{\sqrt{2}}{2}, \tan 300 ^\circ = -\sqrt{3},$ 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

$\sin(a\pm b) &=\sin a \cos b \pm \cos a \sin b \\\cos(a\pm b) &=\cos a \cos b \pm \sin a \sin b \\\tan(a \pm b) &=\frac{\tan a \pm \tan b}{1 \pm \tan a \tan b}$

Example A

Find the exact value of $\sin 75^\circ$ .

Solution: 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 $b = 30^\circ$ .

$\sin 75^\circ &=\sin(45^\circ + 30 ^\circ) \\&= \sin 45^\circ \cos 30^\circ +\cos 45^\circ \sin 30 ^\circ \\&= \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot \frac{1}{2} \\&= \frac{\sqrt{6}+\sqrt{2}}{4}$

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

Example B

Find the exact value of $\cos \frac{11 \pi}{12}$ .

Solution: For this example, we could use either the sum or difference cosine formula, $\frac{11\pi}{12}=\frac{2\pi}{3}+\frac{\pi}{4}$ or $\frac{11\pi}{12}=\frac{7\pi}{6}-\frac{\pi}{4}$ . Let’s use the sum formula.

$\cos \frac{11\pi}{12} &=\cos \left(\frac{2\pi}{3}+\frac{\pi}{4}\right) \\&=\cos \frac{2\pi}{3}\cos \frac{\pi}{4}-\sin\frac{2\pi}{3}\sin \frac{\pi}{4} \\&= -\frac{1}{2}\cdot \frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \\&= -\frac{\sqrt{2}+\sqrt{6}}{4}$

Example C

Find the exact value of $\tan \left(-\frac{\pi}{12}\right)$ .

Solution: This angle is the difference between $\frac{\pi}{4}$ and $\frac{\pi}{3}$ .

$\tan \left(\frac{\pi}{4}-\frac{\pi}{3}\right) &=\frac{\tan \frac{\pi}{4}-\tan \frac{\pi}{3}}{1+\tan \frac{\pi}{4} \tan \frac{\pi}{3}} \\&=\frac{1-\sqrt{3}}{1+\sqrt{3}}$

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

Concept Problem Revisit

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

$\sin 165^\circ &=\sin(120^\circ + 45 ^\circ) \\&= \sin 120^\circ \cos 45^\circ +\cos 120^\circ \sin 45 ^\circ \\&= \frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2}+\frac{-1}{2} \cdot \frac{\sqrt{2}}{2} \\&= \frac{\sqrt{6}-\sqrt{2}}{4}\\$

Guided Practice

Find the exact values of:

1. $\cos 15^\circ$

2. $\tan 255^\circ$

1. $\cos 15^\circ &=\cos(45^\circ - 30^\circ) \\&= \cos 45^\circ \cos 30^\circ - \sin 45 ^\circ \sin 30 ^\circ \\&= \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\cdot \frac{1}{2} \\&= -\frac{\sqrt{6}-\sqrt{2}}{4}$

2. $\tan (210^\circ + 45^\circ) &=\frac{\tan 210^\circ+\tan 45^\circ}{1-\tan 210^\circ \tan 45^\circ} \\&= \frac{\frac{\sqrt{3}}{3}+1}{1-\frac{\sqrt{3}}{3}}=\frac{\frac{\sqrt{3}+3}{3}}{\frac{3-\sqrt{3}}{3}}=\frac{\sqrt{3}+3}{3-\sqrt{3}}$

Vocabulary

Sum and Difference Formulas
$\sin(a\pm b) &=\sin a \cos b \pm \cos a \sin b \\\cos (a \pm b) &=\cos a \cos b \mp \sin a \sin b \\\tan (a \pm b) &=\frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}$

Practice

Find the exact value of the following trig functions.

1. $\sin 15^\circ$
2. $\cos \frac{5\pi}{12}$
3. $\tan 345^\circ$
4. $\cos (-255^\circ)$
5. $\sin \frac{13 \pi}{12}$
6. $\sin \frac{17\pi}{12}$
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 $\tan (a+b)=\frac{\tan a + \tan b}{1-\tan a \tan b}$ .