Math; Trigonometric Identities

[Deleted]

Hello, basicly I'm sitting here doing math and I stumble across a problem I just can't seem to find an answer to without using my calculator (Which I shouldn't have to use.

determine the exact value of cos v if:

sin v=5/13 and v is located in the first quadrant

What I do is just take sin^-1(5/13)=22.62
cos(22.62)=0.923 and I get the answer but the book says no caculator and the answer in the book is cos v=12/13 (Same as cos(22.62))

I would be ever so grateful if you could help me out to understand what it is I'm suppoes to do here to end up with cos v=12/13. I have tried plenty of times without using a calculator but my answers seems to be way of

Thanks

[Deleted]

if v ∈ (π/2,2π) then cos v>0
sin^2v+cos^v=1 fundamental identity of trigonometry
=> cos^2v=1-sin^2v=1-25/169= 144/169
cosv=sqrt(144/169)
covv=+- 12/13 but v ∈(π/2,2π)=> cos v=+12/13

[Badpaladin]

Think about it as a triangle. sin(v) = 5/13 means there's a triangle with the side opposite the angle of length 5, and a hypotenuse of length 13. To get the remaining side's length, you need to apply a^2 + b^2 = c^2, which when manipulated properly will give you your side adjacent to the angle, which when divided by the hypotenuse gives you your cos(v) answer. You can do this without worrying about any positive or negative signs because it's in the first quadrant.

[Blueobelisk]

Think about it as a triangle. sin(v) = 5/13 means there's a triangle with the side opposite the angle of length 5, and a hypotenuse of length 13. To get the remaining side's length, you need to apply a^2 + b^2 = c^2, which when manipulated properly will give you your side adjacent to the angle, which when divided by the hypotenuse gives you your cos(v) answer. You can do this without worrying about any positive or negative signs because it's in the first quadrant.

This guy is completely right. Just draw a triangle, use Pythagorean formula to find the extra side, and put that new side's length over the hypotenuse (13). Done deal.

[Wikiy]

cos^2=1-sin^2
=1-(-5/13)^2
=1-25/169
=144/169
cos=sqrt(144/169)=+-12/13

Since sin is in this case in the first quadrant, so is cos, therefor out of the + and - case we choose the + case.

cos(v)=12/13

[Knight Gil]

My brain is melting

[v2prwsmb45yhuq3wj23vpjk]

Think about it as a triangle. sin(v) = 5/13 means there's a triangle with the side opposite the angle of length 5, and a hypotenuse of length 13. To get the remaining side's length, you need to apply a^2 + b^2 = c^2, which when manipulated properly will give you your side adjacent to the angle, which when divided by the hypotenuse gives you your cos(v) answer. You can do this without worrying about any positive or negative signs because it's in the first quadrant.

It's been about 7 years since I took any sort of trigonometry, but doesn't the Pythagorean Theorem only work with right triangles?

Edit: Looks like it's a right triangle anyway if cos=12/13.

[Wikiy]

It's been about 7 years since I took any sort of trigonometry, but doesn't the Pythagorean Theorem only work with right triangles?

Edit: Looks like it's a right triangle anyway if cos=12/13.

The angle is always right if you take the 3 points as 1 being somewhere on the x axis, one in the centre of the coordinate system and the circle, and 1 being somewhere on the circle itself. Which is how you get sine and cosine anyways, so yeah, it will always be a right triangle in these sorts of tasks.