Question

Find cos θ if sin θ = (-5/13) and tan θ > 0.

Answer 1

please help!

Answer 2

I'll help in a minute :) Just finishing up with another problem.

Answer 3

okay thank you

Answer 4

They told us that \(\large \tan\theta \gt 0\), that tells us that we're either in the 1st or 3rd quadrant.

They also told us that the \(\large \sin\theta\) is negative.
So is our angle theta in the 1st or 3rd quadrant?
We want to be in the quadrant where sine is negative.

Answer 5

Do you understand what I'm asking? :o

Answer 6

first

Answer 7

Hmm, no.
The sine of an angle produces a negative value in the `3rd quadrant`.

Answer 8

oh okay

Answer 9

so where do you go from here?

Answer 10

What we want to do from here is, remember our triangle relationships with the trig functions.

\(\large \sin\theta=\dfrac{-5}{13}=\dfrac{opposite}{hypotenuse}\)

So we'll draw a triangle in the 3rd quadrant, and label the sides.

Answer 11

ok i drew it

Answer 12

From here, do you understand how to label 2 of the sides? Based on the information they gave us about sine.

Answer 13

yes opp=-5
hyp=13

Answer 14

Ok good!

Answer 15

If you're ever confused about where to put the negative sign, with the 5 or the 13, try to remember this little fact.

The hypotenuse should always be written positive.

Answer 16

So from here, we can use the `pythagorean theorem` to solve for that missing side.

Answer 17

okay now what?

Answer 18

\[\large (-5)^2+(adj)^2=13^2\]

Understand how to solve for that missing side?

Answer 19

12

Answer 20

Ok good.
Now here is something we need to be careful about.

Should the 12 be positive or negative?

Think about which way the line is facing.

Answer 21

so the answer would be negative correct?

Answer 22

yes good :)