1. ## Trig Identities

Hey guy, I need help with another problem,

Suppose that $3\pi /2 < \Theta < 2\pi$ and $\sec \Theta = 3$ Evaluate the following

$a. \sin \Theta$
$b. \cos (-\Theta)$
$c. \tan (\pi /2 - \Theta)$
$d. \cot ( \pi + \Theta )$
$e. \csc (2\pi + \Theta )$

2. Originally Posted by goliath
Hey guy, I need help with another problem,

Suppose that $3\pi /2 < \Theta < 2\pi$ and $\sec \Theta = 3$ Evaluate the following

$a. \sin \Theta$
$b. \cos (-\Theta)$
$c. \tan (\pi /2 - \Theta)$
$d. \cot ( \pi + \Theta )$
$e. \csc (2\pi + \Theta )$
The angle is in the third quadrant, where the sine if negative and the cosine is positive. The rest of the tri ratios have signs that depend on the sine and cosine.

Secant is 1/cosine, so cosine(theta) is 1/3.

Use the Trig Identity $\sin^2x + \cos^2 x = 1$ and definitions/properties to find the remaining measures.

Good luck!

3. Hello, goliath!

Suppose that $\tfrac{3\pi}{2} \,<\, \theta \,<\, 2\pi$ and $\sec\theta \,=\, 3$.
$\sec\theta \:=\:\frac{3}{1} \:=\:\frac{hup}{adj}$

We have: . $adj = 1,\;hyp = 3$
And Pythagorus says: . $opp = \pm2\sqrt{2}$

Since $\theta$ is in Quadrant 4, $opp = -2\sqrt{2}$

And we have this list: . $\begin{Bmatrix}\sin\theta &=&-\dfrac{2\sqrt{2}}{3} \\ \\[-3mm] \cos\theta &=& \dfrac{1}{3} \\ \\[-3mm] \tan\theta &=& -2\sqrt{2} \end{Bmatrix}$

Evaluate the following.

$a.\;\sin\theta$
From our list: . $\sin\theta \:=\:-\frac{2\sqrt{2}}{3}$

$b.\;\cos (-\theta)$
$\cos(-\theta) \;=\;\cos\theta \;=\;\frac{1}{3}$

$c.\;\tan \left(\tfrac{\pi}{2} - \theta\right)$
$\tan\left(\tfrac{\pi}{2}-\theta\right) \;=\;\cot\theta \;=\;\frac{1}{\tan\theta} \;=\;-\frac{1}{2\sqrt{2}}$

$d.\;\cot(\pi + \theta)$
$\cot(\pi + \theta) \;=\;\cot\theta \;=\;-\frac{1}{2\sqrt{2}}$

$e.\;\csc(2\pi + \theta )$
$\csc(2\pi + \theta) \;=\;\csc\theta \;=\;\frac{1}{\sin\theta} \;=\;\frac{1}{\text{-}\frac{2\sqrt{2}}{3}} \;=\;-\frac{3}{2\sqrt{2}}$

4. Thank you!, kudos