Question

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a right triangle has side lengths 5, 12, and 13 as shown below. use these lengths to find sinx, tanx, and cosx. sinx = tanx = cosx =

Explanation:

Step1: Recall the definitions of trigonometric functions

In a right - triangle, for an acute angle \(X\), \(\sin X=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\tan X = \frac{\text{opposite}}{\text{adjacent}}\), \(\cos X=\frac{\text{adjacent}}{\text{hypotenuse}}\).
For angle \(X\), the side opposite to \(X\) is \(YZ = 5\), the side adjacent to \(X\) is \(XZ=12\), and the hypotenuse \(XY = 13\).

Step2: Calculate \(\sin X\)

Using the formula \(\sin X=\frac{\text{opposite}}{\text{hypotenuse}}\), we substitute the values: \(\sin X=\frac{5}{13}\).

Step3: Calculate \(\tan X\)

Using the formula \(\tan X=\frac{\text{opposite}}{\text{adjacent}}\), we substitute the values: \(\tan X=\frac{5}{12}\).

Step4: Calculate \(\cos X\)

Using the formula \(\cos X=\frac{\text{adjacent}}{\text{hypotenuse}}\), we substitute the values: \(\cos X=\frac{12}{13}\).

Answer:

\(\sin X=\frac{5}{13}\), \(\tan X=\frac{5}{12}\), \(\cos X=\frac{12}{13}\)