If sin ⁡x=7/25, and 0<x<π/2, what is cos⁡(x−3π/2)?

Question

I need help pleasee...

If sin ⁡x=7/25, and 0<x<π/2, what is cos⁡(x−3π/2)?

Answer 1

Using the Pythagorean Theorem (or recognizing that 24 completes the Pythagorean Triple 7,24, 25) the 2nd leg of the right triangle with an acute angle having a sine with a value of 7/25 is 24. That means the cosine of that angle is 24/25. You can also use the Pythagorean Identity sin²(x) + cos²(x) = 1, which implies that:

(7/25)² + cos²(x) = 1, cos²(x) = 1 - 49/625 = 576/625; cos(x) = ±√(576/625) = ±24/25 (chose positive value because the angle is in the 1st quadrant).

Now use the trig identity:

cos (A - B) = cosA cosB + sinAsinB

cos(x - 3π/2) = cosx cos(3π/2) + sinxsin(3π/2)

= cos(x)(0) + (7/25)(-1)

= -7/25