before-calculus

Question

Find the exact value of sin(A-B) if sinA= 5/13 and sinB = 8/17, A and B are between 0 and pi/2

Answer 1

Remember that 5, 12, 13 form a Pythagorean Triple: 5² + 12² = 13² and form a right triangle.

If sin A = 5 / 13, and A is in the first quadrant, then cos A = 12 / 13.

Remember that 8, 15, and 17 form a Pythagorean Triple: 8² + 15² = 17² and form a right triangle.

If sin B = 8 / 17, and A is in the first quadrant, then cos B = 15 / 17.

Since 5 / 13 < 8 /17, then sin A < sin B.

f(x) = sin(x) is an increasing function in the interval 0 > x > pi/2, hence A < B.

Since A < B, then A - B < 0 and sin (A-B) < 0. Thus, we should get a negative value for sin (A-B):

sin ( A - B ) = sin A cos B - sin B cos A

= (5 / 13) ( 15 / 17) - (8 / 17) (12 / 13)

= (5)(15) / ( 13* 17 ) - ( 8 * 12) / ( 13*17 )

= (75 - 96) / (13 * 17)

= -21 / 221, Ans.