Trig Identities

**goliath** · November 23rd 2009, 11:13 AM

Hey guy, I need help with another problem,

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

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

**apcalculus** · November 23rd 2009, 04:06 PM

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!

**Soroban** · November 23rd 2009, 06:40 PM

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.

$a.\;\sin\theta$