If cosx= -2/3 and pi/2, find the exact value of cos(pi/3-x)
To find the exact value of cos(pi/3 - x), we need to use trigonometric identities and the given information.
First, we can find the value of x using the given information that cos(x) = -2/3. Since the value of cos(x) is negative, we know that x is in the second or third quadrant.
We know that cos(pi/2) = 0, which means that x = pi/2 is not a solution since cos(pi/2) is positive.
Therefore, x must be in the third quadrant. So we can find x by using the inverse cosine function:
x = arccos(-2/3)
Using a calculator, we find that x ≈ 2.3005 radians.
Now, we can substitute this value of x into the expression cos(pi/3 - x):
cos(pi/3 - x) = cos(pi/3 - 2.3005)
To simplify further, we can use the cosine difference identity:
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
Let a = pi/3 and b = 2.3005:
cos(pi/3 - 2.3005) = cos(pi/3)cos(2.3005) + sin(pi/3)sin(2.3005)
We know that cos(pi/3) = 1/2 and sin(pi/3) = sqrt(3)/2, so we can substitute these values:
cos(pi/3 - x) = (1/2)cos(2.3005) + (sqrt(3)/2)sin(2.3005)
We can now calculate the values of cos(2.3005) and sin(2.3005) using a calculator.
After substituting these values into the expression, we get the exact value of cos(pi/3 - x).