Feel pretty dumb, basic trig question

**Peleus** · January 16th 2009, 04:52 AM

If $\sin\theta = \frac{1}{3}$ and $\tan\theta < 0$ then what is $\cos\theta$

Like I said I feel completely dumb here. I know this is so basic I'm just having a mental block with it.

Multiple choice answers are

a) $\frac{1}{3}$

b) $\frac{-3}{\sqrt{10}}$

c) $\frac{-1}{2\sqrt{2}}$

d) $\frac{-2\sqrt{2}}{3}$

e) $\frac{1}{\sqrt{10}}$

Because of the nature of the answers I know I'd have to use the unit triangle somewhere. Because tan is negative and sin is positive the answer is in quadrent 2. In other words sin1/3 = 180 - cos.

What I don't know though is how we've got the unit triangle to give sin 1/3, my stupid question is how do I get sin 1/3 on the unit triangle?

**mathaddict** · January 16th 2009, 05:15 AM

Originally Posted by Peleus

If $\sin\theta = \frac{1}{3}$ and $\tan\theta < 0$ then what is $\cos\theta$

Like I said I feel completely dumb here. I know this is so basic I'm just having a mental block with it.

Multiple choice answers are

a) $\frac{1}{3}$

b) $\frac{-3}{\sqrt{10}}$

c) $\frac{-1}{2\sqrt{2}}$

d) $\frac{-2\sqrt{2}}{3}$

e) $\frac{1}{\sqrt{10}}$

Because of the nature of the answers I know I'd have to use the unit triangle somewhere. Because tan is negative and sin is positive the answer is in quadrent 2. In other words sin1/3 = 180 - cos.

What I don't know though is how we've got the unit triangle to give sin 1/3, my stupid question is how do I get sin 1/3 on the unit triangle?

From the information given , we know that $\theta$ is in the second quadrant . Draw a triangle and find $cos\theta$

**Soroban** · January 16th 2009, 06:11 AM

Hello, Peleus!

If $\sin\theta = \tfrac{1}{3}$ and $\tan\theta < 0$ , find $\cos\theta.$

. . $(a)\;\frac{1}{3}\qquad (b)\;-\frac{3}{\sqrt{10}} \qquad (c)\;-\frac{1}{2\sqrt{2}} \qquad (d)\;-\frac{2\sqrt{2}}{3} \qquad (e)\;\frac{1}{\sqrt{10}}$

Because $\tan\theta < 0\text{ and }\sin\theta > 0$ , the answer is in quadrant 2. . Good!

I don't use the unit circle . . .

We have: . $\sin\theta \:=\:\frac{1}{3} \:=\:\frac{opp}{hyp}$

So $\theta$ is in a right triangle with: . $opp = 1,\;hyp = 3$

Pythagorus says: . $opp^2 + adj^2 \:=\:hyp^2$

So we have: . $1^2 + adj^2 \:=\:3^2 \quad\Rightarrow\quad adj^2 \:=\:8 \quad\Rightarrow\quad adj \:=\:\pm2\sqrt{2}$

Then: . $\cos\theta \:=\:\frac{adj}{hyp} \:=\:\pm\frac{2\sqrt{2}}{3}$

In quadrant 2, cosine is negative: . ${\color{blue}\cos\theta\:=\:-\frac{2\sqrt{2}}{3}}\quad\hdots$ answer (d)

**Peleus** · January 16th 2009, 12:27 PM

It's times like this that the answer smacks you in the face and tells you how silly you are.

Thank you very much and quite obvious now. I guess I just have to not get focused on looking at one method to solve it (unit circle) and look at the big picture.

Thanks again.

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