Question

If cos ⁡x= −4/5 , and π < x < 3π/2, what is sin⁡(x + π/2)?Enter your answer as a fraction in simplest form, like this: 3/14

Answer 1

sin(x+π/2) = -4/5

Step-by-step explanation:

To find sin(x+π/2), we can use the identity sin(a+b) = sin(a)cos(b) + cos(a)sin(b). In this case, a = x and b = π/2. We are given that cos(x) = -4/5. Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can find sin(x) = √(1 - cos^2(x)). Since π < x < 3π/2, sin(x) is negative. Therefore, sin(x) = -√(1 - (-4/5)^2) = -√(1 - 16/25) = -√(9/25) = -3/5.

Now, we can substitute these values into the identity sin(a+b) = sin(a)cos(b) + cos(a)sin(b) to find sin(x+π/2). sin(x+π/2) = sin(x)cos(π/2) + cos(x)sin(π/2) = (-3/5)(0) + (-4/5)(1) = -4/5.