Question

question 2 (multiple choice worth 1 points) (05.02 mc) in △jkl, solve for x. image of right triangle jkl with right angle at k, kl = 34, angle at l is 27°, hypotenuse jl = x options: 74.89, 30.29, 38.16, 66.73

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \( \triangle JKL \), we know the adjacent side to the \( 27^\circ \) angle (\( KL = 34 \)) and we need to find the hypotenuse \( x \) (JL). The cosine function relates the adjacent side and the hypotenuse: \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \).

Step2: Substitute the values into the cosine formula

We have \( \theta = 27^\circ \), adjacent \( = 34 \), and hypotenuse \( = x \). So, \( \cos(27^\circ)=\frac{34}{x} \).

Step3: Solve for \( x \)

Rearrange the formula to solve for \( x \): \( x=\frac{34}{\cos(27^\circ)} \). Calculate \( \cos(27^\circ)\approx0.8910 \), then \( x = \frac{34}{0.8910}\approx38.16 \). Wait, no, wait—wait, actually, in the triangle, \( KL \) is adjacent to angle \( L \), and \( JL \) is the hypotenuse? Wait, no, wait: angle at \( L \) is \( 27^\circ \), right angle at \( K \). So side \( KL = 34 \) (adjacent to angle \( L \)), side \( JK \) is opposite, and \( JL \) is hypotenuse. Wait, but if we use cosine, \( \cos(27^\circ)=\frac{KL}{JL}=\frac{34}{x} \), so \( x=\frac{34}{\cos(27^\circ)} \). Let's recalculate: \( \cos(27^\circ)\approx0.8910 \), so \( 34\div0.8910\approx38.16 \)? Wait, but wait, maybe I mixed up. Wait, no, wait: wait, maybe it's the adjacent and hypotenuse? Wait, no, let's check again. Wait, angle at \( L \) is \( 27^\circ \), right angle at \( K \). So sides: \( KL \) is adjacent to angle \( L \), \( JK \) is opposite, \( JL \) is hypotenuse. So cosine of angle \( L \) is adjacent over hypotenuse, so \( \cos(27^\circ)=\frac{KL}{JL} \), so \( JL = \frac{KL}{\cos(27^\circ)} = \frac{34}{\cos(27^\circ)} \approx \frac{34}{0.8910} \approx 38.16 \). Wait, but let's check the options. One of the options is 38.16. Wait, but let's confirm the calculation. \( \cos(27^\circ) \approx 0.8910 \), so \( 34 / 0.8910 \approx 38.16 \).

Answer:

38.16