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answer attempt 1 out of 3 sin z = cos y = sin z and cos y

Question

answer attempt 1 out of 3
sin z =
cos y =
sin z and cos y

Explanation:

Step1: Find the length of XY (opposite to Z)

First, we know it's a right triangle with hypotenuse ZY = 51 and one leg XZ = 45. Using the Pythagorean theorem \(a^2 + b^2 = c^2\), where \(c = 51\), \(b = 45\), and \(a = XY\). So \(XY^2 + 45^2 = 51^2\). Calculate \(51^2 = 2601\) and \(45^2 = 2025\). Then \(XY^2 = 2601 - 2025 = 576\), so \(XY = \sqrt{576} = 24\).

Step2: Calculate \(\sin Z\)

In a right triangle, \(\sin Z=\frac{\text{opposite}}{\text{hypotenuse}}\). For angle Z, the opposite side is XY = 24 and the hypotenuse is ZY = 51. So \(\sin Z=\frac{24}{51}=\frac{8}{17}\) (simplify by dividing numerator and denominator by 3).

Step3: Calculate \(\cos Y\)

In a right triangle, \(\cos Y=\frac{\text{adjacent}}{\text{hypotenuse}}\). For angle Y, the adjacent side is XY (wait, no, let's check the triangle. Angle Y: the right angle is at X, so sides: XY, XZ, ZY. Angle Y: adjacent side is XY? Wait, no. Wait, angle Y: the sides: XY is opposite to Z, XZ is opposite to Y? Wait, no. Wait, in triangle XYZ, right-angled at X. So angle at Y: the sides: XY is one leg, XZ is the other leg, ZY is hypotenuse. So for angle Y, the adjacent side is XY? Wait, no. Wait, angle Y: the sides: the side adjacent to Y is XY? Wait, no, let's label the triangle: X is right angle, so XZ and XY are legs, ZY is hypotenuse. So angle at Z: between XZ and ZY. Angle at Y: between XY and ZY. So for angle Y, the adjacent side is XY, and the hypotenuse is ZY. Wait, no, adjacent side to Y is XY? Wait, no, adjacent side is the leg that forms angle Y with the hypotenuse. So angle Y is at vertex Y, so the sides forming angle Y are YX and YZ. So the adjacent side is YX (length 24), and the hypotenuse is YZ (length 51). Wait, no, wait: in right triangle, for angle Y, the adjacent side is the leg that is part of angle Y, excluding the hypotenuse. So angle Y is between YX and YZ. So YX is a leg (length 24), YZ is hypotenuse (51), and XZ is the other leg (45). Wait, no, XZ is length 45, XY is length 24. So for angle Y, the adjacent side is XY (length 24)? No, wait, no: in angle Y, the two sides are YX (from Y to X) and YZ (from Y to Z). The right angle is at X, so YX is perpendicular to XZ. So angle Y: the adjacent side is YX (length 24), and the opposite side is XZ (length 45). Wait, no, cosine of angle Y is adjacent over hypotenuse. Adjacent side to Y is the side that is part of angle Y and is a leg (not hypotenuse). So angle Y is at Y, so the sides are YX (leg) and YZ (hypotenuse), and XZ (leg). So the adjacent side to Y is YX (length 24), and hypotenuse is YZ (51). Wait, but let's check: \(\cos Y = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{XY}{YZ}\)? Wait, no, that can't be. Wait, maybe I mixed up. Wait, in right triangle XYZ, right-angled at X. So:

  • For angle Z:
  • Opposite side: XY (length 24)
  • Adjacent side: XZ (length 45)
  • Hypotenuse: ZY (length 51)
  • For angle Y:
  • Opposite side: XZ (length 45)
  • Adjacent side: XY (length 24)
  • Hypotenuse: ZY (length 51)

Wait, no, that's not right. Wait, angle at Y: the sides: from Y to X (XY, length 24), from Y to Z (YZ, length 51), and from X to Z (XZ, length 45). So angle Y is between XY and YZ. So the adjacent side to Y is XY (length 24), and the opposite side is XZ (length 45). So \(\cos Y = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{XY}{YZ} = \frac{24}{51} = \frac{8}{17}\)? Wait, no, that's the same as \(\sin Z\). Wait, let's check: \(\sin Z = \frac{XY}{YZ} = \frac{24}{51} = \frac{8}{17}\), and \(\cos Y = \frac{XY}{YZ} = \frac{24}{51} = \frac{8}{17}\)? Wait, that can't be. Wait, no, maybe I got…

Answer:

\(\sin Z = \frac{8}{17}\), \(\cos Y = \frac{8}{17}\), and \(\sin Z\) and \(\cos Y\) are equal.