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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 tan3π8\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 (2a)\begin{align*}(2a)\end{align*} and Half-Angle (a2)\begin{align*}\left(\frac{a}{2}\right)\end{align*} Formulas.

#### Double-Angle and Half-Angle Formulas

cos2asina2cosa2=cos2asin2a=2cos2a1=1sin2a=±1cosa2=±1+cosa2sin2a=2sinacosatan2a=2tana1tan2atana2=1cosasina  =sina1+cosa\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 sina2\begin{align*}\sin \frac{a}{2}\end{align*} and cosa2\begin{align*}\cos \frac{a}{2}\end{align*} depend on which quadrant a2\begin{align*}\frac{a}{2}\end{align*} lies in. For cos2a\begin{align*}\cos 2a\end{align*} and tana2\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 cosπ8\begin{align*}\cos \frac{\pi}{8}\end{align*}.

π8\begin{align*}\frac{\pi}{8}\end{align*} is half of π4\begin{align*}\frac{\pi}{4}\end{align*} and in the first quadrant.

cos(12π4)=1+cosπ42=1+222=122+22=2+22\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*}

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

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