Verify cos(x-3pi/2)=-sin x
3 pi / 2 = 270 °
sin ( 3 pi / 2 ) = sin 270 ° = - 1
cos ( 3 pi / 2 ) = cos 270 ° = 0
cos ( A - B ) = cos A cos B + sin A sin B
In this case :
cos ( x - 3 pi / 2 ) = cos ( x ) * cos ( 3 pi / 2 ) + sin ( x ) * sin ( 3 pi / 2 ) =
cos ( x ) * 0 + sin ( x ) * ( - 1 ) =
0 - sin ( x ) = - sin ( x )
To verify the equality cos(x - 3π/2) = -sin(x), we can make use of the trigonometric identity that relates the cosine and sine functions.
The identity we will use is:
cos(α - β) = cosα * cosβ + sinα * sinβ
In this case, let's substitute α = x and β = 3π/2 into the identity:
cos(x - 3π/2) = cos(x) * cos(3π/2) + sin(x) * sin(3π/2)
Now, we need to determine the cosine and sine values of 3π/2. Recall that the cosine function is negative and the sine function is positive at this angle due to its location in the third quadrant of the unit circle.
cos(3π/2) = 0
sin(3π/2) = -1
Now, let's continue simplifying the expression:
cos(x - 3π/2) = cos(x) * 0 + sin(x) * (-1)
cos(x - 3π/2) = -sin(x)
Therefore, the intermediate step confirms that cos(x - 3π/2) is indeed equal to -sin(x).