Question

find the principal solution of the equation tan5theta = -1
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Answer 1

Answer:

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Answer 2

Final Answer:

The principal solution of the equation $tan5\theta = -1$ is $\theta=\frac{7\pi}{20} or\frac{3\pi}{20}$.

Given:

The equation $tan5\theta = -1$

To Find:

The principal solution of the equation $tan5\theta = -1$.

Explanation:

Note the following essential points.

• The trigonometric term $tan\theta$ is equal to the division of $sin\theta$ by $cos\theta$.

• The value of the trigonometric term $tan\theta$ is positive in the first quadrant and in the third quadrant.

• The value of the trigonometric term $tan\theta$ is negative in the second quadrant and in the fourth quadrant.

• The value of the trigonometric term $tan\frac{\pi}{4}$ is 1.
• The value of the trigonometric term $-tan\frac{\pi}{4}$ is -1.

Step 1 of 3

Considering the second quadrant, evaluate the following.

$tan5\theta\\= -tan\frac{\pi}{4}\\= tan(-\frac{\pi}{4})\\= tan(\pi-\frac{\pi}{4})\\=tan(\frac{3\pi}{4})$

Step 2 of 3

Considering the fourth quadrant, evaluate the following.

$tan5\theta\\= -tan\frac{\pi}{4}\\= tan(-\frac{\pi}{4})\\= tan(2\pi-\frac{\pi}{4})\\=tan(\frac{7\pi}{4})$

Step 3 of 3

Thus, the principal solution of the equation $tan5\theta = -1$ is found out in the following way.

$5\theta=\frac{7\pi}{4} or\frac{3\pi}{4}\\\theta=\frac{7\pi}{20} or\frac{3\pi}{20}$

Therefore, the required principal solution of the equation $tan5\theta = -1$ is $\theta=\frac{7\pi}{20} or\frac{3\pi}{20}$.

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