An acute angle \theta satisfies the equation \sin(\theta) = \frac{24}{25}. What is the val… — SAT

An acute angle $\theta$ satisfies the equation $\sin(\theta) = \frac{24}{25}$ . What is the value of $\frac{1}{\cos(\theta)} - \tan(\theta)$ ?

The correct value of the expression is

\frac{1}{7}

The correct answer is

\frac{1}{7}

. Using the Pythagorean identity, we determine that

\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \frac{7}{25}

. Using the definition of tangent, we find

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{24}{7}

. Substituting these values into the given expression yields

\frac{1}{7/25} - \frac{24}{7} = \frac{25}{7} - \frac{24}{7} = \frac{1}{7}

Find the value of

\cos(\theta)

using the Pythagorean identity

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

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

Since

\theta

is an acute angle,

\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{24}{25}\right)^2} = \sqrt{\frac{49}{625}} = \frac{7}{25}

Find the value of

\tan(\theta)

using the definition

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

\tan(\theta) = \frac{24}{7}

Substituting the known values gives

\tan(\theta) = \frac{24/25}{7/25} = \frac{24}{7}

Substitute the values of

\cos(\theta)

and

\tan(\theta)

into the expression

\frac{1}{\cos(\theta)} - \tan(\theta)

and simplify.

\frac{1}{7}

The expression becomes

\frac{1}{7/25} - \frac{24}{7} = \frac{25}{7} - \frac{24}{7} = \frac{1}{7}

Pythagorean identity and trigonometric ratio definitions in a right triangle

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