If \sin(\theta) = \frac{\sqrt{5}}{3} and \theta is an acute angle, what is the value of \t… — SAT

If $\sin(\theta) = \frac{\sqrt{5}}{3}$ and $\theta$ is an acute angle, what is the value of $\tan^2(\theta)$ ?

Answer: 1.25

The correct answer is 1.25 (or the equivalent fraction 5/4).

Using the Pythagorean identity

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

, we find

\cos^2(\theta) = 1 - \frac{5}{9} = \frac{4}{9}

. Since

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

, it follows that

\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)} = \frac{5/9}{4/9} = 1.25

Calculate the square of the cosine of the angle using the Pythagorean identity.

\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \left(\frac{\sqrt{5}}{3}\right)^2 = 1 - \frac{5}{9} = \frac{4}{9}

The Pythagorean identity states that

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

for any angle

\theta

Express the tangent squared in terms of sine squared and cosine squared.

\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)}

By definition, the tangent function is the ratio of sine to cosine, so

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

Substitute the values of

\sin^2(\theta)

and

\cos^2(\theta)

to solve for

\tan^2(\theta)

\tan^2(\theta) = \frac{5/9}{4/9} = \frac{5}{4} = 1.25

Plugging in the squared ratios and simplifying gives the final evaluation.

Pythagorean trigonometric identity and quotient identity of tangent

Question