Question

Find tanθ if sinθ=−25√5 is in the third quadrant.

Answer 1

Using a trigonometric identity, it is found that the tangent of the angle is of:

$\tan{\theta} = \frac{2}{11}$

How to find the sine of an angle given the cosine, or vice versa?

The sine and the cosine of the angle are related according to the following identity:

$\sin^2{\theta} + \cos^2{\theta} = 1$

For this problem, the sine is given by:

$\sin{\theta} = -\frac{2}{5\sqrt{5}}$

Hence the cosine is found as follows:

$\left(-\frac{2}{5\sqrt{5}}\right)^2 + \cos^2{\theta} = 1$

$\frac{4}{125} + \cos^2{\theta} = 1$

$\cos^2{\theta} = \frac{121}{125}$

$\cos{\theta} = \pm \sqrt{\frac{121}{125}}$

On the third quadrant the cosine is negative, hence:

$\cos{\theta} = -\frac{11}{5\sqrt{5}}$

What is the tangent of an angle?

The tangent of an angle is given by the sine divided by the cosine, hence:

$\tan{\theta} = \frac{-\frac{2}{5\sqrt{5}}}{-\frac{11}{5\sqrt{5}}} = \frac{2}{11}$

More can be learned about trigonometric identities at brainly.com/question/26676095

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