Question

If sin Θ = 2 over 3 and tan Θ < 0, what is the value of cos Θ?

Answer 1

Hey! Do you know in which quadrant \(\sin\theta\) will be positive and \(\tan\theta\) will be negative?

Answer 2

yes, sin theta is positive when the y coordinate is positive, and tan is negative when either sin or cos is negative. i'm just thinking either the second or the last one is right

Answer 3

That's good! And yes! We're able to eliminate those two answers because we know that cos has to be negative.

Answer 4

um so is it the second or the fourth... both has values of a negative number

Answer 5

Have you learned about the trigonometric identity

\(\sin ^2\theta + \cos^2 \theta = 1\)

Answer 6

no? what's that

Answer 7

It's called a trigonometric identity but basically it's like equations that involve trig functions that are always true

Answer 8

And so we can use that to solve for \(\cos \theta\)

Answer 9

uh, how

Answer 10

we know that \(\sin\theta = \dfrac{2}{3}\)

\(\sin^2\theta = \left(\dfrac{2}{3} \right)^2\)

What is \(\left(\dfrac{2}{3} \right)^2 = ?\)

Answer 11

4/9

Answer 12

\(\sin ^2\theta + \cos^2 \theta = 1\)

So now we get

\(\dfrac{4}{9} + \cos^2 \theta = 1\)

What is 1 - 4/9?

And let's try to keep it as a fraction

Answer 13

5/9

Answer 14

Good!

So now we get

\(\cos^2\theta = \dfrac{5}{9}\)

and that's \(\cos^2 \theta\) and we want \(\cos\theta\) so we take the square root on both sides!

Does that make sense? So

\(\cos\theta = \sqrt{\dfrac{5}{9}}\)

Can you simplify that?


Remember \(\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}\)

Answer 15

it would be sqrt5/sqrt9, simplify- sqrt5/3

Answer 16

oh

Answer 17

ohhhhhhhh, i see, thxxxxxxxxxxxx

Answer 18

Good!

So we know it's sqrt(5)/3

But remember the first thing we mentioned? \(\cos\theta\) has to be negative so it'll be -sqrt(5)/3

Answer 19

oh, right lol, well, thxx

Answer 20

Of course!!! It was my pleasure :)

Answer 21

lol

Answer 22

\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
Good!

So we know it's sqrt(5)/3

But remember the first thing we mentioned? \(\cos\theta\) has to be negative so it'll be -sqrt(5)/3
\(\color{#0cbb34}{\text{End of Quote}}\)

the - got stuck in the previous line
but
\( - \dfrac{\sqrt{5}}{3}\)

Answer 23

thx

Answer 24

You're welcome!