Question

Express sin(xπ/2) and cos(xπ/2) in terms of sinx and cosx:

a) sin(xπ/2) = cos(x) and cos(xπ/2) = sin(x)
b) sin(xπ/2) = sin(x) and cos(xπ/2) = cos(x)
c) sin(xπ/2) = cos(x) and cos(xπ/2) = -sin(x)
d) sin(xπ/2) = -sin(x) and cos(xπ/2) = cos(x)

Answer 1

The correct expressions for sin(xπ/2) and cos(xπ/2) are sin(xπ/2) = cos(x) and cos(xπ/2) = -sin(x) based on co-function identities in trigonometry.

Step-by-step explanation:

The expression of sin(xπ/2) and cos(xπ/2) in terms of sin(x) and cos(x) requires understanding of the unit circle and trigonometric identities. In particular, the angle xπ/2 corresponds to a rotation from the x-axis by that angle in the unit circle. The correct relationship uses the co-function identities, where the sine of an angle is equal to the cosine of its complement and vice versa, which gives us the following correspondences:

• sin(xπ/2) = cos(x)
• cos(xπ/2) = -sin(x)

Therefore, the correct option is (c): sin(xπ/2) = cos(x) and cos(xπ/2) = -sin(x).