Question

If cos Θ = square root 2 over 2 and 3 pi over 2 < Θ < 2π, what are the values of sin Θ and tan Θ ?

Answer 1

@dunnitagainn

Answer 2

Wait, I may be wrong, but isn't the " pi over 2 < Θ < 2π" part irrelevant to finding the sin and tan of this problem?

Answer 3

yeah, for the most part, i just tells you what direction the triangle is facing.

Answer 4

I'm just drawing it out right now, have you learned about the unit circle yet?

Answer 5

Yeah, I've had a few lessons on it, but I only understand like 40% of it. I know about the counter-clockwise, quadrants, and degreees to radians.

Answer 6

so basically you would use the x and y coordinates of the graph, the x is equal to cos and y is equal to sin, so since the cos is square root of 2 over 2, on the unit circle, the y coordinate is also square root of 2 over two. (this is for the first quadrant only, it changes signs based on which quadrant you are in). so you now know the cos and sin those number are your a and b for the c^2 = a^2+b^2, so C^2= (sqrt2/2)^2+(sqrt2/2)^2

Answer 7

does that make sense so far?

Answer 8

It makes sense so far.

Answer 9

then when you do your math, you would get 1, which is the tangent. (this makes sense when your cos and sin are the same which happens when you have the sqrt2/2 for both)

Answer 10

but what you get for c is not always the tangent, this is only when your cos and sin are equal, because when you put them over one another, obviously you get 1.

Answer 11

Cos= Adj/Hyp
I'm given: Cos= Sqr2/2

Adj side= Sqr2
Hyp= 2

Now, I use pythagorean Identity.


\[\sqrt{2}^2+b^2=2^2\

The power of "^2" cancels out the sqrt and leaves: 2
Then, the \[c^2\] (2^2) = 4

\[2+b^2=4\]

Now simplify...?

Subtract 2 from both sides

\[b^2=2\]

\[b=\sqrt{2}\]

Answer 12

??

Answer 13

When you are dealing with the numbers from the unit circle, you do things a little differently, but you aren't breaking any rules, the x value is cos theta and the y value is sin theta. This means that the ordered pair in this case would be (sort2/2, sqrt2/2) so that means that those are your cos and sin values. you do not break these up, in order to find tan you can just put x over y, so you would have (sqrt2/2)/(sqrt2/2) which is equal to 1. does that make sense?

Answer 14

Yeah, that makes sense.

Answer 15

okay, and there are some complex mathematical rules to why you can do that, but honestly i never learned them and they don't really matter as long as you know that x=cos theta, y=sin theta, and x/y=tan theta

Answer 16

Thanks, I'll make sure to write those in my notes!

We would be in the fourth quadrant right? If so, doesn't that mean one or both of the quadrants would be negative??

Answer 17

yeah you are right sorry, it is in the 4th quadrant which will make the sin negative since y is negative in that quadrant which also makes the tangent negative. Sorry i overlooked that

Answer 18

Thanks, I solved it. Can you still help with a few more??