Expert Answer to "Use half-angle identities to find an exact value..." - Plainmath

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Mabel Breault

Answered question

2022-01-16

Use half-angle identities to find an exact value without a calculator.

{\tan 195}^{\circ}

Answer & Explanation

vrangett

Beginner2022-01-17Added 36 answers

Let

\frac{u}{2} = 195^{\circ}, t h e n u = 390^{\circ}

. We can reduce u to

30^{\circ}

by subtracting

360^{\circ}

. Now use one of the tan half angle identities

\tan \frac{u}{2} = \frac{1 - \cos u}{\sin u}

{\tan 195}^{\circ} = \frac{1 - {\cos 30}^{\circ}}{{\sin 30}^{\circ}} = \frac{1 - \frac{\sqrt{3}}{2}}{\frac{1}{2}} = 2 - \sqrt{3}

Result:

2 - \sqrt{3}

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