Question

Sin theta =-(1)/(\sqrt(17))and (3\pi )/(2)<=\theta <=2\pi then tan\theta =

Answer 1

Using a trigonometric identity, and considering that the angle is in the fourth quadrant, the tangent of the angle is given as follows:

tan(theta) = -1/4

Which trigonometric identity relates the sine and the cosine of an angle?

The following identity is used to relate the measures, considering an angle $\theta$:

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

For this problem, the sine is given as follows:

$\sin{\theta} = -\frac{1}{\sqrt{17}}$

Then the cosine, which we need to find the tangent, is found as follows:

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

$\left(-\frac{1}{\sqrt{17}}\right)^2 + \cos^2{\theta} = 1$

$\frac{1}{17} + \cos^2{\theta} = 1$

$\cos^2{\theta} = \frac{16}{17}$

$\cos{\theta} = \pm \sqrt{\frac{16}{17}}$

Since the angle is in the fourth quadrant, the cosine is positive, hence:

$\cos{\theta} = \frac{4}{\sqrt{17}}$

What is the tangent of an angle?

It is the sine of the angle divided by the cosine of the angle, hence:

$\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} = \frac{-\frac{1}{\sqrt{17}}}{\frac{4}{\sqrt{17}}} = -\frac{1}{4}$

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

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