3.3.7: Finding Exact Trigonometric Values Using Sum and Difference Formulas

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.

2. \(\cos \dfrac{11 \pi}{12}\)

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.

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

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

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 \(\tan (a+b)=\dfrac{\tan a+\tan b}{1−\tan a\tan b}\).

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 14.12.

Additional Resources

Video: Finding Exact Values of Trigonometric Functions Using Sum and Difference Formulas - Overview

Practice: Finding Exact Trigonometric Values Using Sum and Difference Formulas