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Math  /  Trigonometry

Question−tan⁡(19π3)-\tan \left(\frac{19 \pi}{3}\right)

Studdy Solution

STEP 1

1. The tangent function, tan⁡(θ)\tan(\theta), is periodic with a period of π\pi, meaning tan⁡(θ+kπ)=tan⁡(θ)\tan(\theta + k\pi) = \tan(\theta) for any integer kk.
2. The angle 19π3\frac{19\pi}{3} can be simplified by finding an equivalent angle within the principal range [0,2π)[0, 2\pi).
3. The tangent function is odd, meaning tan⁡(−θ)=−tan⁡(θ)\tan(-\theta) = -\tan(\theta).

STEP 2

1. Simplify the angle 19π3\frac{19\pi}{3} to an equivalent angle within [0,2π)[0, 2\pi).
2. Calculate the tangent of the simplified angle.
3. Apply the negative sign to the result.

STEP 3

First, simplify the angle 19π3\frac{19\pi}{3} by finding its equivalent angle within [0,2π)[0, 2\pi).
Since 2π=6π32\pi = \frac{6\pi}{3}, subtract multiples of 2π2\pi from 19π3\frac{19\pi}{3}:
19π3=19π3−6π=19π3−18π3=π3 \frac{19\pi}{3} = \frac{19\pi}{3} - 6\pi = \frac{19\pi}{3} - \frac{18\pi}{3} = \frac{\pi}{3}
So, 19π3\frac{19\pi}{3} is equivalent to π3\frac{\pi}{3}.

STEP 4

Calculate the tangent of the simplified angle π3\frac{\pi}{3}:
tan⁡(π3)=3 \tan\left(\frac{\pi}{3}\right) = \sqrt{3}

STEP 5

Apply the negative sign to the result:
−tan⁡(19π3)=−3 -\tan\left(\frac{19\pi}{3}\right) = -\sqrt{3}
The solution is:
−tan⁡(19π3)=−3 -\tan\left(\frac{19\pi}{3}\right) = -\sqrt{3}

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