In right triangle XYZ, the measure of angle Y is 90^\circ, XY = 12, and YZ = 5. What is th… — SAT

In right triangle $XYZ$ , the measure of angle $Y$ is $90^\circ$ , $XY = 12$ , and $YZ = 5$ . What is the value of $\cos(X) - \sin(X)$ ?

\frac{7}{13}

The length of the hypotenuse

XZ

is first found using the Pythagorean theorem:

XZ = \sqrt{12^2 + 5^2} = 13

. Using the definitions of the trigonometric ratios,

\cos(X) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}

and

\sin(X) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}

. The difference is

\cos(X) - \sin(X) = \frac{12}{13} - \frac{5}{13} = \frac{7}{13}

Find the length of the hypotenuse

XZ

using the Pythagorean theorem.

XZ = \sqrt{XY^2 + YZ^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13

The hypotenuse is needed as the denominator for the sine and cosine ratios.

Determine the values of

\cos(X)

and

\sin(X)

\cos(X) = \frac{XY}{XZ} = \frac{12}{13}

and

\sin(X) = \frac{YZ}{XZ} = \frac{5}{13}

By definition,

\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

and

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

Calculate the difference

\cos(X) - \sin(X)

\cos(X) - \sin(X) = \frac{12}{13} - \frac{5}{13} = \frac{7}{13}

This is the final subtraction requested by the question.

Calculating trigonometric ratios in a right triangle and applying basic operations.

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