Sum and Difference Identities Math Example 3

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Example 3

medium
Find the exact value of tan⁡(15°)\tan(15°) using a difference identity.

Solution

  1. 1
    Write 15°=45°−30°15° = 45° - 30° and apply tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}.
  2. 2
    tan⁡(15°)=tan⁡45°−tan⁡30°1+tan⁡45°⋅tan⁡30°=1−331+33=3−333+33=3−33+3\tan(15°) = \frac{\tan 45° - \tan 30°}{1 + \tan 45° \cdot \tan 30°} = \frac{1 - \frac{\sqrt{3}}{3}}{1 + \frac{\sqrt{3}}{3}} = \frac{\frac{3-\sqrt{3}}{3}}{\frac{3+\sqrt{3}}{3}} = \frac{3-\sqrt{3}}{3+\sqrt{3}}.
  3. 3
    Rationalize: (3−3)2(3+3)(3−3)=9−63+39−3=12−636=2−3\frac{(3-\sqrt{3})^2}{(3+\sqrt{3})(3-\sqrt{3})} = \frac{9 - 6\sqrt{3} + 3}{9-3} = \frac{12 - 6\sqrt{3}}{6} = 2 - \sqrt{3}.

Answer

tan⁡(15°)=2−3\tan(15°) = 2 - \sqrt{3}
The tangent difference formula tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} allows us to compute exact tangent values for non-standard angles by decomposing them into known angles. Rationalizing the denominator simplifies the result.

About Sum and Difference Identities

Formulas that express sin⁡(A±B)\sin(A \pm B), cos⁡(A±B)\cos(A \pm B), and tan⁡(A±B)\tan(A \pm B) in terms of sin⁡A\sin A, cos⁡A\cos A, sin⁡B\sin B, and cos⁡B\cos B.

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More Sum and Difference Identities Examples

Example 1 easy

Find the exact value of [formula] using the sum identity.

Example 2 medium

Simplify [formula].

Example 4 hard

If [formula] with [formula] in QI and [formula] with [formula] in QII, find [formula].

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