Question

we know angle l has a measure of 77° and side jl has length 20. which trig function should we use to find x? ○ sin 77° ○ cos 77° ○ tan 77°

Explanation:

To determine the trigonometric function, we analyze the right triangle (implied by angle \( L = 77^\circ \) and side \( JL = 20 \)). Let's assume \( x \) is the opposite side to \( \angle L \), and \( JL \) is the adjacent side (or vice versa, but the key is the ratio).

• Sine (\( \sin \theta \)) is \( \frac{\text{opposite}}{\text{hypotenuse}} \), but we don't know the hypotenuse here.
• Cosine (\( \cos \theta \)) is \( \frac{\text{adjacent}}{\text{hypotenuse}} \), also not directly helpful if \( x \) is opposite.
• Tangent (\( \tan \theta \)) is \( \frac{\text{opposite}}{\text{adjacent}} \). If \( JL \) is adjacent and \( x \) is opposite to \( \angle L \), then \( \tan 77^\circ=\frac{x}{JL} \), so we use \( \tan 77^\circ \) to solve for \( x \) (since \( JL = 20 \), we can rearrange \( x = JL \cdot \tan 77^\circ \)). Wait, no—wait, maybe I mixed up. Wait, actually, let's correct: If \( JL \) is the adjacent side, and \( x \) is the opposite side, then \( \tan \theta=\frac{\text{opposite}}{\text{adjacent}} \), so \( \tan 77^\circ=\frac{x}{JL} \), so we use \( \tan 77^\circ \). Wait, but maybe the original problem has \( JL \) as the adjacent and \( x \) as opposite. Alternatively, if \( JL \) is the hypotenuse, but no—wait, the options are sin, cos, tan. Let's re-express:

Wait, maybe the triangle is labeled with \( J \), \( L \), \( K \), right-angled at \( K \). So \( \angle L = 77^\circ \), side \( JL \) is a leg (adjacent or opposite). Wait, no—actually, the key is that tangent is opposite over adjacent, which doesn't require the hypotenuse. So if we have the adjacent side (JL = 20) and we need the opposite side (x), then \( \tan 77^\circ=\frac{x}{20} \), so we use \( \tan 77^\circ \).

Wait, but maybe I made a mistake. Wait, no—let's think again. The question is "which trig function to use to find x". So if \( x \) is opposite, and \( JL \) is adjacent, then tan is the ratio of opposite over adjacent. So the correct function is \( \tan 77^\circ \).

Wait, no—wait, maybe I had it backwards. Wait, maybe \( JL \) is the hypotenuse? No, the options are sin, cos, tan. Let's recall:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

If we need to find \( x \), and we know an angle and a side, and we want to relate the opposite and adjacent sides, we use tangent. So the correct trig function is \( \tan 77^\circ \).

Wait, but wait—maybe I messed up. Wait, no—let's check the options. The options are sin 77°, cos 77°, tan 77°. So the answer is tan 77°.

Wait, no—wait, maybe the original problem has \( JL \) as the adjacent side, and \( x \) as the opposite side. So \( \tan 77^\circ = \frac{x}{JL} \), so we use \( \tan 77^\circ \) to solve for \( x \). So the correct option is \( \tan 77^\circ \).

Answer:

\( \tan 77^\circ \) (the option: \( \tan 77^\circ \))