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Finding Exact Trigonometric Values Using Sum and Difference Formulas

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

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Finding Exact Trig Values using Sum and Difference Formulas

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

Sum and Difference Formulas

You know that \begin{align*}\sin 30^\circ=\frac{1}{2}, \cos 135^\circ=-\frac{\sqrt{2}}{2}, \tan 300 ^\circ = -\sqrt{3},\end{align*} 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 \begin{align*}30^\circ, 45^\circ,\end{align*} and \begin{align*}60^\circ\end{align*}. Using the Sum and Difference Formulas, we can find these exact trig values.

Sum and Difference Formulas

\begin{align*}\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}\end{align*}

Find the exact value using the Sum and Difference Formulas

Find the exact value of \begin{align*}\sin 75^\circ\end{align*}.

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

\begin{align*}\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}\end{align*}

In general, \begin{align*}\sin (a+b)\ne \sin a+\sin b\end{align*} and similar statements can be made for the other sum and difference formulas.

Find the exact value of \begin{align*}\cos \frac{11 \pi}{12}\end{align*}.

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

\begin{align*}\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}\end{align*}

Find the exact value of \begin{align*}\tan \left(-\frac{\pi}{12}\right)\end{align*}.

This angle is the difference between \begin{align*}\frac{\pi}{4}\end{align*} and \begin{align*}\frac{\pi}{3}\end{align*}.

\begin{align*}\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}}\end{align*}

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

Examples

Example 1

Earlier, you were asked what is the exact sine of an angle without using the calculator. 

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

\begin{align*}\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}\\\end{align*}

Example 2

Find the exact value of:

\begin{align*}\cos 15^\circ\end{align*}
\begin{align*}\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}\end{align*}

Example 3

Find the exact value of: 

\begin{align*}\tan 255^\circ\end{align*}


\begin{align*}\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}}\end{align*}

Review

Find the exact value of the following trig functions.

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

Answers for Review Problems

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

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