If \theta is an acute angle such that \sin \theta = \frac{5}{13}, what is the exact numeri… — JAMB UTME

If $\theta$ is an acute angle such that $\sin \theta = \frac{5}{13}$ , what is the exact numerical value of $169(\sin^2 \theta - \cos^2 \theta)$ ?

Cevap: -119

The exact numerical value of the expression is

-119

Using the Pythagorean identity

\sin^2 \theta + \cos^2 \theta = 1

, we find

\cos^2 \theta = 1 - \frac{25}{169} = \frac{144}{169}

. Then

\sin^2 \theta - \cos^2 \theta = \frac{25 - 144}{169} = -\frac{119}{169}

. Multiplying by

169

gives

-119

Calculate

\cos^2 \theta

using the Pythagorean trigonometric identity.

\cos^2 \theta = \frac{144}{169}

Since

\sin^2 \theta + \cos^2 \theta = 1

, subtracting

\sin^2 \theta = \frac{25}{169}

from

1

yields

\frac{144}{169}

Compute the difference

\sin^2 \theta - \cos^2 \theta

\sin^2 \theta - \cos^2 \theta = -\frac{119}{169}

Subtracting

\frac{144}{169}

from

\frac{25}{169}

gives

-\frac{119}{169}

Scale the difference by

169

169 \times \left(-\frac{119}{169}\right) = -119

Multiplying

-\frac{119}{169}

169

cancels the denominator, leaving

-119

Pythagorean Trigonometric Identity

Soru