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Finding Exact Trigonometric Values Using Double Angle Identities

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Finding Exact Trig Values using Double and Half Angle Formulas

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

Guidance

In the previous concept, we added two different angles together to find the exact values of trig functions. 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) and Half-Angle \left(\frac{a}{2}\right) Formulas.

Double-Angle and Half-Angle Formulas

\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}

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

Example A

Find the exact value of \cos \frac{\pi}{8} .

Solution: \frac{\pi}{8} is half of \frac{\pi}{4} and in the first quadrant.

\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}

Example B

Find the exact value of \sin 2a if \cos a=- \frac{4}{5} and \frac{3 \pi}{2} \le a < 2 \pi .

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

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

Example C

Find the exact value of \tan 2a for a from Example B.

Solution: Use \tan a=\frac{\sin a}{\cos a}=\frac{\frac{3}{5}}{- \frac{4}{5}}=- \frac{3}{4} to solve for \tan 2a .

\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}

Concept Problem Revisit

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

\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}}}

If we simplify this expression, we get \sqrt{2} + 1 .

Guided Practice

1. Find the exact value of \cos \left(-\frac{5 \pi}{8}\right) .

2. \cos a=\frac{4}{7} and 0 \le a < \frac{\pi}{2} . Find:

a) \sin 2a

b) \tan \frac{a}{2}

Answers

1. - \frac{5 \pi}{8} is in the 3^{rd} quadrant.

- \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}

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

a) \sin 2a=2 \cdot \frac{\sqrt{33}}{7} \cdot \frac{4}{7}=\frac{8 \sqrt{33}}{49}

b) You can use either \tan \frac{a}{2} formula.

\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}

Vocabulary

Double-Angle and Half-Angle Formulas
\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}

Practice

Find the exact value of the following angles.

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

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

  1. \sin 2a
  2. \cos \frac{a}{2}
  3. \tan \frac{a}{2}
  4. \cos 2a

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

  1. \tan 2a
  2. \sin \frac{a}{2}
  3. \cos \frac{a}{2}
  4. \sin 2a

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