For an acute angle \theta, \cos(\theta) = \frac{5}{13}. What is the value of 5\tan(\theta)… — SAT

For an acute angle $\theta$ , $\cos(\theta) = \frac{5}{13}$ . What is the value of $5\tan(\theta) + 13\sin(\theta)$ ?

Answer: 24

Using the Pythagorean identity

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

with

\cos(\theta) = \frac{5}{13}

gives

\sin(\theta) = \frac{12}{13}

because

\theta

is an acute angle. The quotient identity gives

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{12}{5}

. Substituting these ratios into the expression

5\tan(\theta) + 13\sin(\theta)

gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 12 + 12 = 24

Find the value of

\sin(\theta)

using the Pythagorean identity.

\sin(\theta) = \frac{12}{13}

Since

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

and

\theta

is an acute angle, the sine value is positive:

\sin(\theta) = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \frac{12}{13}

Find the value of

\tan(\theta)

using the quotient identity.

\tan(\theta) = \frac{12}{5}

By definition,

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

. Substituting the known values yields

\tan(\theta) = \frac{12/13}{5/13} = \frac{12}{5}

Substitute the trigonometric ratios into the given expression and simplify.

Substituting the values of

\tan(\theta)

and

\sin(\theta)

into

5\tan(\theta) + 13\sin(\theta)

gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 12 + 12 = 24

Trigonometric ratios and identities, specifically the Pythagorean identity

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

and the definition of tangent as

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

Alternatively, draw a right triangle with an acute angle

\theta

. Since

\cos(\theta) = \frac{5}{13} = \frac{\text{adjacent}}{\text{hypotenuse}}

, label the adjacent side as 5 and the hypotenuse as 13. By the Pythagorean theorem, the opposite side is

\sqrt{13^2 - 5^2} = 12

. From this triangle,

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13}

and

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5}

. Substituting these values into the expression gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 24

Estimated Time:1m 30s

Question