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Finding Exact Trigonometric Values Using Double Angle Identities

Sine, cosine, and tangent of angles other than multiples of 30, 45, and 60 degrees.

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Finding Exact Trig Values using Double and Half Angle Formulas

You want to find the exact value of \begin{align*}\tan \frac{3\pi}{8}\end{align*}. How could you find this value without using a calculator?

Double Angle and Half Angle Formulas

In this concept, we will learn how to find the exact values of the trig functions for angles that are half or double of other angles. Here we will introduce the Double-Angle \begin{align*}(2a)\end{align*} and Half-Angle \begin{align*}\left(\frac{a}{2}\right)\end{align*} Formulas.

Double-Angle and Half-Angle Formulas

\begin{align*}\cos 2a&=\cos^2 a- \sin^2a && \sin 2a=2 \sin a \cos a \\ &=2 \cos^2 a-1 && \tan 2a=\frac{2 \tan a}{1- \tan^2 a} \\ &=1- \sin^2a \\ \sin \frac{a}{2}&= \pm \sqrt{\frac{1- \cos a}{2}} && \tan \frac{a}{2}=\frac{1- \cos a}{\sin a} \\ \cos \frac{a}{2}&= \pm \sqrt{\frac{1+ \cos a}{2}} && \qquad \ \ =\frac{\sin a}{1+ \cos a}\end{align*}

The signs of \begin{align*}\sin \frac{a}{2}\end{align*} and \begin{align*}\cos \frac{a}{2}\end{align*} depend on which quadrant \begin{align*}\frac{a}{2}\end{align*} lies in. For \begin{align*}\cos 2a\end{align*} and \begin{align*}\tan \frac{a}{2}\end{align*} any formula can be used to solve for the exact value.

Let's find the exact value of \begin{align*}\cos \frac{\pi}{8}\end{align*}.

\begin{align*}\frac{\pi}{8}\end{align*} is half of \begin{align*}\frac{\pi}{4}\end{align*} and in the first quadrant.

\begin{align*}\cos \left(\frac{1}{2} \cdot \frac{\pi}{4}\right)&=\sqrt{\frac{1+ \cos \frac{\pi}{4}}{2}} \\ &=\sqrt{\frac{1+ \frac{\sqrt{2}}{2}}{2}} \\ &=\sqrt{\frac{1}{2} \cdot \frac{2+\sqrt{2}}{2}} \\ &=\frac{\sqrt{2+ \sqrt{2}}}{2}\end{align*}

Now, let's find the exact value of \begin{align*}\sin 2a\end{align*} if \begin{align*}\cos a=- \frac{4}{5}\end{align*} and \begin{align*}\frac{3 \pi}{2} \le a < 2 \pi\end{align*}.

To use the sine double-angle formula, we also need to find \begin{align*}\sin a\end{align*}, which would be \begin{align*}\frac{3}{5}\end{align*} because \begin{align*}a\end{align*} is in the \begin{align*}4^{th}\end{align*} quadrant.

\begin{align*}\sin 2a&=2 \sin a \cos a \\ &=2 \cdot \frac{3}{5} \cdot - \frac{4}{5} \\ &=- \frac{24}{25}\end{align*}

Finally, let's find the exact value of \begin{align*}\tan 2a\end{align*} for \begin{align*}a\end{align*} from the previous problem.

Use \begin{align*}\tan a=\frac{\sin a}{\cos a}=\frac{\frac{3}{5}}{- \frac{4}{5}}=- \frac{3}{4}\end{align*} to solve for \begin{align*}\tan 2a\end{align*}.

\begin{align*}\tan 2a=\frac{2 \cdot - \frac{3}{4}}{1- \left(- \frac{3}{4}\right)^2}=\frac{- \frac{3}{2}}{\frac{7}{16}}= - \frac{3}{2} \cdot \frac{16}{7}=- \frac{24}{7}\end{align*}

Examples

Example 1

Earlier, you were asked to find the value of \begin{align*}\tan \frac{3\pi}{8}\end{align*} without a calculator. 

\begin{align*}\frac{3\pi}{8} = \frac{1}{2} \cdot \frac{3\pi}{4}\end{align*} so we can use the formula \begin{align*}tan \frac{a}{2} =\frac{\sin a}{1+ \cos a}\end{align*} for \begin{align*}a=\frac{3\pi}{4}\end{align*}

\begin{align*}\tan \frac{3\pi}{8} =\frac {\sin \frac{3\pi}{4}}{1+ \cos \frac{3\pi}{4}}\\ =\frac {\frac{\sqrt{2}}{2}}{{1 +\frac{-\sqrt{2}}{2}}}\end{align*}

If we simplify this expression, we get \begin{align*}\sqrt{2} + 1\end{align*}.

Example 2

Find the exact value of \begin{align*}\cos \left(-\frac{5 \pi}{8}\right)\end{align*}.

\begin{align*}- \frac{5 \pi}{8}\end{align*} is in the \begin{align*}3^{rd}\end{align*} quadrant.

\begin{align*}- \frac{5 \pi}{8}=\frac{1}{2} \left(- \frac{5 \pi}{4}\right) \rightarrow \cos \frac{1}{2} \left(- \frac{5 \pi}{4}\right)=- \sqrt{\frac{1+ \cos \left(- \frac{5 \pi}{4}\right)}{2}}=-\sqrt{\frac{1- \frac{\sqrt{2}}{2}}{2}}=\sqrt{\frac{1}{2} \cdot \frac{2-\sqrt{2}}{2}}=\frac{\sqrt{2- \sqrt{2}}}{2}\end{align*}

Example 3

Given the function \begin{align*}\cos a=\frac{4}{7}\end{align*} and \begin{align*}0 \le a < \frac{\pi}{2}\end{align*}, find \begin{align*}\sin 2a\end{align*}.

First, find \begin{align*}\sin a\end{align*}. \begin{align*}4^2+y^2=7^2\rightarrow y=\sqrt{33}\end{align*}, so \begin{align*}\sin a=\frac{\sqrt{33}}{7}\end{align*}

\begin{align*}\sin 2a=2 \cdot \frac{\sqrt{33}}{7} \cdot \frac{4}{7}=\frac{8 \sqrt{33}}{49}\end{align*}

Example 4

Given the function \begin{align*}\cos a=\frac{4}{7}\end{align*} and \begin{align*}0 \le a < \frac{\pi}{2}\end{align*}, find \begin{align*}\tan \frac{a}{2}\end{align*}.

You can use either \begin{align*}\tan \frac{a}{2}\end{align*} formula.

\begin{align*}\tan \frac{a}{2}=\frac{1- \frac{4}{7}}{\frac{\sqrt{33}}{7}}=\frac{3}{7} \cdot \frac{7}{\sqrt{33}}=\frac{3}{\sqrt{33}}=\frac{\sqrt{33}}{11}\end{align*}

Review

Find the exact value of the following angles.

  1. \begin{align*}\sin 105^\circ\end{align*}
  2. \begin{align*}\tan \frac{\pi}{8}\end{align*}
  3. \begin{align*}\cos \frac{5 \pi}{12}\end{align*}
  4. \begin{align*}\cos 165^\circ\end{align*}
  5. \begin{align*}\sin \frac{3 \pi}{8}\end{align*}
  6. \begin{align*}\tan \left(- \frac{\pi}{12}\right)\end{align*}
  7. \begin{align*}\sin \frac{11 \pi}{8}\end{align*}
  8. \begin{align*}\cos \frac{19 \pi}{12}\end{align*}

The \begin{align*}\cos a= \frac{5}{13}\end{align*} and \begin{align*}\frac{3 \pi}{2} \le a < 2 \pi\end{align*}. Find:

  1. \begin{align*}\sin 2a\end{align*}
  2. \begin{align*}\cos \frac{a}{2}\end{align*}
  3. \begin{align*}\tan \frac{a}{2}\end{align*}
  4. \begin{align*}\cos 2a\end{align*}

The \begin{align*}\sin a=\frac{8}{11}\end{align*} and \begin{align*}\frac{\pi}{2} \le a < \pi\end{align*}. Find:

  1. \begin{align*}\tan 2a\end{align*}
  2. \begin{align*}\sin \frac{a}{2}\end{align*}
  3. \begin{align*}\cos \frac{a}{2}\end{align*}
  4. \begin{align*}\sin 2a\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 14.15. 

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