Use identities to find the exact value of the trigonometric function. Find sin alpha given that cos alpha = 4/7 and 3pi/2 < alpha < 2pi

Answer 1

We know that the Pythagorean identity is: sin^2(alpha) + cos^2(alpha) = 1.

Given that cos(alpha) = 4/7, we can substitute this value into the Pythagorean identity to solve for sin(alpha):

sin^2(alpha) + (4/7)^2 = 1
sin^2(alpha) + 16/49 = 1
sin^2(alpha) = 1 - 16/49
sin^2(alpha) = 33/49

Taking the square root of both sides, we get:

sin(alpha) = ±sqrt(33/49)

Since 3pi/2 < alpha < 2pi, alpha is in the fourth quadrant where sin(alpha) is negative. Therefore,

sin(alpha) = -sqrt(33/49)

Hence, sin(alpha) = -sqrt(33)/7.