Use identities to find the exact value of the trigonometric function. Find sun alpha given that cos alpha = 4/7
First, let's use the Pythagorean identity to find the value of sin alpha.
The Pythagorean identity states that sin^2(alpha) + cos^2(alpha) = 1.
We already know that cos alpha = 4/7, so we can substitute that in:
sin^2(alpha) + (4/7)^2 = 1
sin^2(alpha) + 16/49 = 1
sin^2(alpha) = 1 - 16/49
sin^2(alpha) = 33/49
Now, take the square root of both sides to find sin alpha:
sin(alpha) = ± sqrt(33/49)
Since sin alpha is positive in the first and second quadrants of the unit circle, we can take the positive square root:
sin(alpha) = sqrt(33/49)
Now we have the values of both sin alpha and cos alpha:
sin(alpha) = sqrt(33/49)
cos(alpha) = 4/7