If \cos(\theta) = \frac{7}{25} and 0 < \theta < \frac{\pi}{2}, what is the value of \sin\l… — SAT

If $\cos(\theta) = \frac{7}{25}$ and $0 < \theta < \frac{\pi}{2}$ , what is the value of $\sin\left(\frac{\pi}{2} - \theta\right)$ ?

\frac{7}{25}

The correct answer is

\frac{7}{25}

because the co-function identity states that for any angle

\theta

, the sine of its complement,

\frac{\pi}{2} - \theta

, is equal to the cosine of

\theta

. Given that

\cos(\theta) = \frac{7}{25}

, it follows that

\sin\left(\frac{\pi}{2} - \theta\right) = \frac{7}{25}

Identify the relevant trigonometric identity.

Use the co-function identity

\sin\left(\frac{\pi}{2} - x\right) = \cos(x)

Since the angle is given in radians, the complementary angle identity relates the sine of the complement to the cosine of the original angle.

Substitute the given value.

Since

\cos(\theta) = \frac{7}{25}

, then

\sin\left(\frac{\pi}{2} - \theta\right) = \frac{7}{25}

Direct substitution of the given cosine value into the identity yields the final answer.

Co-function identities relate the sine and cosine of complementary angles.

Question