Question

which trigonometric ratios are correct for triangle xyz? check all that apply. tan(y) = 8/15, cos(x) = 15/17, tan(x) = 15/8, sin(y) = 8/17, cos(y) = 8/17. (image shows right triangle xyz with right angle at z, zx = 8, zy = 15)

Explanation:

First, we find the hypotenuse \( XY \) using the Pythagorean theorem. In right triangle \( XYZ \) with legs \( XZ = 8 \) and \( YZ = 15 \), the hypotenuse \( XY \) is:

Step1: Calculate hypotenuse \( XY \)

$$ XY=\sqrt{8^{2}+15^{2}}=\sqrt{64 + 225}=\sqrt{289}=17 $$

Step2: Analyze \( \tan(Y) \)

For angle \( Y \), the opposite side is \( XZ = 8 \) and the adjacent side is \( YZ = 15 \). So \( \tan(Y)=\frac{\text{opposite}}{\text{adjacent}}=\frac{8}{15} \), so this is correct.

Step3: Analyze \( \cos(X) \)

For angle \( X \), the adjacent side is \( XZ = 8 \)? Wait, no. Wait, angle \( X \): the adjacent side is \( XZ = 8 \)? Wait, no. Wait, in right triangle at \( Z \), angle \( X \): the adjacent side is \( XZ = 8 \)? Wait, no. Wait, angle \( X \): the sides: opposite is \( YZ = 15 \), adjacent is \( XZ = 8 \), hypotenuse is \( XY = 17 \). Wait, no, \( \cos(X)=\frac{\text{adjacent}}{\text{hypotenuse}} \). The adjacent side to angle \( X \) is \( XZ = 8 \)? Wait, no, I made a mistake. Wait, angle \( X \): the sides: the leg adjacent to \( X \) is \( XZ = 8 \), and hypotenuse is \( XY = 17 \)? Wait, no, no. Wait, in triangle \( XYZ \), right - angled at \( Z \), so:

• For angle \( X \):
• Opposite side: \( YZ = 15 \)
• Adjacent side: \( XZ = 8 \)
• Hypotenuse: \( XY = 17 \)

So \( \cos(X)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{8}{17} \)? Wait, no, the option is \( \cos(X)=\frac{15}{17} \). Wait, I messed up. Wait, no, angle \( X \): adjacent side is \( XZ = 8 \)? Wait, no, no. Wait, the right angle is at \( Z \), so sides:

• \( XZ = 8 \) (one leg, adjacent to \( X \) and \( Z \))
• \( YZ = 15 \) (the other leg, adjacent to \( Y \) and \( Z \))
• \( XY = 17 \) (hypotenuse)

So for angle \( X \):

• \( \cos(X)=\frac{\text{adjacent to }X}{\text{hypotenuse}}=\frac{XZ}{XY}=\frac{8}{17} \)? But the option is \( \cos(X)=\frac{15}{17} \). Wait, no, I think I mixed up opposite and adjacent. Wait, angle \( X \): the side opposite to \( X \) is \( YZ = 15 \), the side adjacent to \( X \) is \( XZ = 8 \), hypotenuse is \( XY = 17 \). Wait, no, \( \cos(X)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{8}{17} \), but the option is \( \cos(X)=\frac{15}{17} \). Wait, maybe I made a mistake. Wait, no, let's re - check. Wait, the option is \( \cos(X)=\frac{15}{17} \). Wait, maybe I got the angle wrong. Wait, no, let's check the other options.

Step4: Analyze \( \tan(X) \)

For angle \( X \), \( \tan(X)=\frac{\text{opposite}}{\text{adjacent}}=\frac{YZ}{XZ}=\frac{15}{8} \), so this is correct.

Step5: Analyze \( \sin(Y) \)

For angle \( Y \), the opposite side is \( XZ = 8 \), hypotenuse is \( XY = 17 \). So \( \sin(Y)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{8}{17} \), so this is correct.

Step6: Analyze \( \cos(Y) \)

For angle \( Y \), the adjacent side is \( YZ = 15 \), hypotenuse is \( XY = 17 \). So \( \cos(Y)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{15}{17} \), but the option is \( \cos(Y)=\frac{8}{17} \), so this is incorrect.

Answer:

The correct trigonometric ratios are:

• \( \tan(Y)=\frac{8}{15} \)
• \( \tan(X)=\frac{15}{8} \)
• \( \sin(Y)=\frac{8}{17} \)
• \( \cos(X)=\frac{8}{17} \) is wrong, so the correct options are \( \tan(Y)=\frac{8}{15} \), \( \tan(X)=\frac{15}{8} \), \( \sin(Y)=\frac{8}{17} \) (and also \( \tan(Y)=\frac{8}{15} \) was correct, \( \tan(X)=\frac{15}{8} \) is correct, \( \sin(Y)=\frac{8}{17} \) is correct, and \( \tan(Y)=\frac{8}{15} \) is correct. Wait, let's re - summarize:

1. \( \tan(Y)=\frac{8}{15} \): Correct (opposite = 8, adjacent = 15 for angle \( Y \))
2. \( \cos(X)=\frac{15}{17} \): Incorrect (adjacent to \( X \) is 8, so \( \cos(X)=\frac{8}{17} \))
3. \( \tan(X)=\frac{15}{8} \): Correct (opposite = 15, adjacent = 8 for angle \( X \))
4. \( \sin(Y)=\frac{8}{17} \): Correct (opposite = 8, hypotenuse = 17 for angle \( Y \))
5. \( \cos(Y)=\frac{8}{17} \): Incorrect (adjacent to \( Y \) is 15, so \( \cos(Y)=\frac{15}{17} \))

So the correct options are \( \tan(Y)=\frac{8}{15} \), \( \tan(X)=\frac{15}{8} \), \( \sin(Y)=\frac{8}{17} \)

In boxed form (but as per instruction, we list the options with their text):

• \( \tan(Y)=\frac{8}{15} \)
• \( \tan(X)=\frac{15}{8} \)
• \( \sin(Y)=\frac{8}{17} \)