Question

the sail of a boat is in the shape of a right triangle. which expression shows the height, in meters, of the sail? image of a right triangle with a 35° angle and a side labeled 8 m options: $\frac{cos 35^circ}{8}$, $8(\tan 35^circ)$, $\frac{\tan 35^circ}{8}$, $8(cos 35^circ)$

Explanation:

Step1: Identify the trigonometric relationship

In a right triangle, the sine, cosine, and tangent functions relate the angles to the sides. The height (opposite side to the 35° angle) and the hypotenuse (8 m) or adjacent side? Wait, no, let's check the triangle. Wait, the given side is 8 m, which is the adjacent side to the 35° angle? Wait, no, maybe the 8 m is the hypotenuse? Wait, no, let's recall: in a right triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Wait, the height is the opposite side to the 35° angle. Let's assume the 8 m is the adjacent side? Wait, no, maybe the 8 m is the hypotenuse? Wait, no, the options: let's look at the options. The options include \(8(\tan35^\circ)\), \(8(\cos35^\circ)\), etc. Wait, maybe the 8 m is the adjacent side, and the height is the opposite side. So \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so \(\text{opposite}=\text{adjacent}\times\tan\theta\). If the adjacent side is 8 m, then height (opposite) is \(8\times\tan35^\circ\)? Wait, no, wait the options: one of the options is \(8(\tan35^\circ)\). Wait, let's re-examine. Wait, the triangle is a right triangle, with angle 35°, the side given is 8 m (let's say it's the adjacent side to the 35° angle), and the height is the opposite side. So \(\tan(35^\circ)=\frac{\text{height}}{8}\), so \(\text{height}=8\times\tan(35^\circ)\)? Wait, no, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so if adjacent is 8, then opposite (height) is \(8\times\tan(35^\circ)\). Wait, but let's check the options. The options are:

1. \(\frac{\cos35^\circ}{8}\)
2. \(8(\tan35^\circ)\)
3. \(\frac{\tan35^\circ}{8}\)
4. \(8(\cos35^\circ)\)

Wait, no, the original options (from the image) are:

• \(\frac{\cos35^\circ}{8}\)
• \(8(\tan35^\circ)\)
• \(\frac{\tan35^\circ}{8}\)
• \(8(\cos35^\circ)\)

Wait, no, the user's image shows:

First option: \(\frac{\cos35^\circ}{8}\)

Second: \(8(\tan35^\circ)\)

Third: \(\frac{\tan35^\circ}{8}\)

Fourth: \(8(\cos35^\circ)\)

Wait, let's correct. Let's assume the right triangle has angle 35°, the side adjacent to 35° is 8 m? No, wait, maybe the 8 m is the hypotenuse? No, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so if hypotenuse is 8, adjacent is \(8\cos\theta\), but that's not height. Wait, height is opposite. So \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), so opposite (height) is \(8\sin35^\circ\), but that's not an option. Wait, maybe the 8 m is the adjacent side, and the height is opposite. So \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so opposite = adjacent × tanθ. If adjacent is 8, then opposite (height) is \(8\tan35^\circ\), which is the second option. So that's the correct expression.

Step2: Verify the trigonometric ratio

In a right triangle, for an angle \(\theta\):

• \(\tan\theta = \frac{\text{opposite side}}{\text{adjacent side}}\)

Let the height (opposite to \(35^\circ\)) be \(h\), and the adjacent side to \(35^\circ\) be \(8\) m (assuming the 8 m is the adjacent side). Then:

\(\tan(35^\circ) = \frac{h}{8}\)

Solving for \(h\):

\(h = 8 \times \tan(35^\circ)\)

Answer:

\(8(\tan 35^\circ)\) (corresponding to the second option, e.g., if options are labeled, it would be the option with \(8(\tan 35^\circ)\))