SOLUTION: 1. In triangle ABC which is right angled at B sin A = A/A+B a) is it reasonable to state that the hypotenuse measures ( A+B) units, justify b) what is the value of 1) cos A , 2

Algebra ->  Trigonometry-basics -> SOLUTION: 1. In triangle ABC which is right angled at B sin A = A/A+B a) is it reasonable to state that the hypotenuse measures ( A+B) units, justify b) what is the value of 1) cos A , 2      Log On


   

Question 826149: 1. In triangle ABC which is right angled at B sin A = A/A+B
a) is it reasonable to state that the hypotenuse measures ( A+B) units, justify
b) what is the value of 1) cos A , 2) tan A 3) cot C
Thanks so much in advance:)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
a) Since sin is opposite over adjacent one would think that if the sin was A%2F%28A%2BB%29, then the hypotenuse would be A+B. But it might not be. The fraction A%2F%28A%2BB%29 might be the reduced form of the opposite/hypotenuse ratio. For example, the opposite side and the hypotenuse might be 2A and 2(A+B) or 14A and 14(A+B), etc.

b) To find cos(A) use
cos%5E2%28A%29+=+1+-+sin%5E2%28A%29
cos%5E2%28A%29+=+1+-+%28A%2F%28A%2BB%29%29%5E2
cos%5E2%28A%29+=+1+-+A%5E2%2F%28A%2BB%29%5E2
cos%5E2%28A%29+=+%28A%2BB%29%5E2%2F%28A%2BB%29%5E2+-+A%5E2%2F%28A%2BB%29%5E2
cos%5E2%28A%29+=+%28A%5E2%2B2AB%2BB%5E2%29%2F%28A%2BB%29%5E2+-+A%5E2%2F%28A%2BB%29%5E2
cos%5E2%28A%29+=+%282AB%2BB%5E2%29%2F%28A%2BB%29%5E2
sqrt%28cos%5E2%28A%29%29+=+sqrt%28%282AB%2BB%5E2%29%2F%28A%2BB%29%5E2%29
We will ignore the negative square root because all the trig functions are positive for any angle that is in a triangle (i.e. an angle from 0 to 90).
cos%28A%29+=+sqrt%28%282AB%2BB%5E2%29%29%2Fsqrt%28%28A%2BB%29%5E2%29
cos%28A%29+=+sqrt%28%282AB%2BB%5E2%29%29%2F%28A%2BB%29

For tan(A) use sin(A)/cos(A):
tan%28A%29+=+%28A%2F%28A%2BB%29%29%2F%28sqrt%282AB%2BB%5E2%29%2F%28A%2BB%29%29
The two "little" denominators cancel:
tan%28A%29+=+A%2Fsqrt%282AB%2BB%5E2%29
I'll leave it up to you to rationalize the denominator.

Since B is the right angle, then A and C are the two acute angles in the triangle. The two acute triangles of a right triangle always add up to 90 degrees. IOW, they are always complementary. We will use this fact for cot(C). Since that tan and cot of complimentary angles are always equal:
cot(C) = tan(A)
So cot(C) will have the same answer as tan(A) (above).