SOLUTION: Tan y =2 find cos y

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Question 1203568: Tan y =2 find cos y
Found 3 solutions by ikleyn, greenestamps, MathLover1:
Answer by ikleyn(52915) About Me  (Show Source):
You can put this solution on YOUR website!
.
tan(y) =2. Find cos(y).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Having given tan(y) = 2, we can imagine a right-angled triangle with acute angle "y", 
the adjacent leg of the length 1 and the opposite leg of the length 2.


Then the hypotenuse length is  sqrt%281%5E2+%2B+2%5E2%29 = sqrt%285%29  and cos(y) = 1%2Fsqrt%285%29 = sqrt%285%29%2F5.   ANSWER

Solved.



Answer by greenestamps(13216) About Me  (Show Source):
You can put this solution on YOUR website!


I'm changing the angle name from y to theta so I can discuss the problem in terms of x, y, and the magnitude r.

Given tan%28theta%29=2, there are two possibilities:

(1) the angle is in quadrant I with y=2, x=1 and r=sqrt%282%5E2%2B1%5E2%29=sqrt%285%29. In this case, cos%28theta%29=x%2Fr=1%2Fsqrt%285%29

(2) the angle is in quadrant III with y=-2, x=-1 and r=sqrt%28%28-2%29%5E2%2B%28-1%291%5E2%29=sqrt%285%29. In this case, cos%28theta%29=x%2Fr=-1%2Fsqrt%285%29

ANSWERS: cos%28theta%29=1%2Fsqrt%285%29 or cos%28theta%29=-1%2Fsqrt%285%29

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

tan%28+y%29+=2
find cos%28+y%29
use trigonometric identities
cos+%28y%29+=+1%2Fsec%28y%29
Pythagorean trigonometric identity
1%2Btan%5E2%28y%29++=+sec%5E2%28y%29
1%2B2%5E2++=+sec%5E2%28y%29
5++=+sec%5E2%28y%29
sec%28y%29=sqrt%285%29
then
cos+%28y%29+=+1%2Fsec%28y%29
cos+%28y%29+=1%2Fsqrt%285%29
cos+%28y%29+=sqrt%285%29%2F5