# Thread: [SOLVED] Find Cos (theta), sin(theta), and tan(theta)

1. ## [SOLVED] Find Cos (theta), sin(theta), and tan(theta)

The problem states,

If sec(theta) = 3 with theta in QIV find cos(theta), sin(theta) and tan(theta).

2. ## Re:

$\sec\theta=3$

$\cos\theta=\frac{1}{3}$

$\theta$ is in the fourth quadrant so cos is positive .

$\sin\theta=-\frac{\sqrt{8}}{3}$

$
\tan\theta=-\sqrt{8}
$

3. Thank you for helping
Can you explain in a little more detail how you got those answers?

4. you should know that secant is the reciprocal of cosine , so the fact that

$\cos{\theta} = \frac{1}{3}$ should be immediate.

since $\theta$ is in quad IV , you should also know that $\sin{\theta}$ and $\tan{theta}$ are both negative values.

there are two ways to determine the values of sine and tangent ...

1) use of reference triangles

sketch a reference right triangle in quad IV ... since $\cos{\theta} = \frac{1}{3}$, then the adjacent side = 1 and the hypotenuse = 3

the size of the opposite side = $\sqrt{3^2 - 1^2} = \sqrt{8} = 2\sqrt{2}$

since the opposite side has a downward direction, its value is $-2\sqrt{2}$.

$\sin{\theta} = \frac{opp}{hyp} = -\frac{2\sqrt{2}}{3}$

$\tan{\theta} = \frac{opp}{adj} = -\frac{2\sqrt{2}}{1} = -2\sqrt{2}$

2) use of Pythagorean identities ...

$\sin{\theta} = \pm \sqrt{1 - \cos^2{\theta}}$

$\tan{\theta} = \pm \sqrt{\sec^2{\theta} - 1}$

and, of course, you have to determine the correct sign from the quadrant info.

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### value of cos theta sin theta and tan theta

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