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Find the exact value of (15^).

March 30, 2026Mathematics

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Answer

2 - \sqrt{3}

4 steps

Here is the solution for number 10.

10. Find the exact value of $\tan(15^\circ)$ .

Express $15^\circ$ as a difference of standard angles.
$\tan(15^\circ) = \tan(45^\circ - 30^\circ)$
Apply the tangent difference formula $\tan(A-B) = \frac{\tan(A)-\tan(B)}{1+\tan(A)\tan(B)}$ .
$= \frac{\tan(45^\circ) - \tan(30^\circ)}{1 + \tan(45^\circ)\tan(30^\circ)}$ $= \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}}$
Simplify the expression.
$= \frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{\sqrt{3}+1}{\sqrt{3}}}$ $= \frac{\sqrt{3}-1}{\sqrt{3}+1}$
Rationalize the denominator.
$= \frac{\sqrt{3}-1}{\sqrt{3}+1} \cdot \frac{\sqrt{3}-1}{\sqrt{3}-1}$ $= \frac{(\sqrt{3})^2 - 2\sqrt{3} + 1^2}{(\sqrt{3})^2 - 1^2}$ $= \frac{3 - 2\sqrt{3} + 1}{3 - 1}$ $= \frac{4 - 2\sqrt{3}}{2}$ $= 2 - \sqrt{3}$
The final answer is $2 - \sqrt{3}$ .