In triangle ACB, where angle C is the right angle, angle B measures \( g^\circ \), angle A measures \( h^\circ \), segment AC measures \( x \), segment CB measures \( y \), and segment AB measures \( z \):
The relationships among the sides and angles in this right triangle can be described using trigonometric functions:
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The sine function relates the opposite side to the hypotenuse. For angle B (which is \( g^\circ \)), the opposite side is AC (which measures \( x \)), and the hypotenuse is AB (which measures \( z \)).
- So, the correct equation for sine is: \[ \sin g^\circ = \frac{x}{z} \]
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The cosine function relates the adjacent side to the hypotenuse, but since we're not asked about that, we'll leave it out.
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The tangent function relates the opposite side to the adjacent side. For angle B (which is \( g^\circ \)), the opposite side is AC (which measures \( x \)), and the adjacent side is CB (which measures \( y \)).
- So, the correct equation for tangent is: \[ \tan g^\circ = \frac{x}{y} \]
From the given options, the correct equation involving the sides for angle B using the sine function is: \[ \sin g^\circ = \frac{x}{z} \]
However, this doesn't appear in your provided options. To clarify, the valid options based on the standard definitions are:
- \( \sin g^\circ = \frac{x}{z} \) (not provided),
- \( \tan g^\circ = \frac{x}{y} \) (not provided).
In your listed equations, the closest is:
- \( \sin g^\circ = z ÷ x \) which is incorrect because it should be \( x ÷ z \).
Thus, none of the listed options are correct based on standard definitions. If we look at the equations specifically for \( z \) (the hypotenuse):
- The closest valid expression would be \( \sin g^\circ = \frac{x}{z} \), but this is not listed in your options.
Therefore, based on provided options, no equation is strictly correct.
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