Question

a right triangle has side lengths 8, 15, and 17 as shown below. use these lengths to find tan m, cos m, and sin m. image of right triangle with right angle at n, sides: mn=8, nl=15, ml=17, vertices m, n, l

Explanation:

Step1: Identify sides for angle M

In right triangle \( \triangle MNL \) with right angle at \( N \), for angle \( M \):

• Opposite side (to \( M \)): \( NL = 15 \)
• Adjacent side (to \( M \)): \( MN = 8 \)
• Hypotenuse: \( ML = 17 \)

Step2: Calculate \( \tan M \)

\( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \), so \( \tan M = \frac{NL}{MN} = \frac{15}{8} \)

Step3: Calculate \( \cos M \)

\( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \), so \( \cos M = \frac{MN}{ML} = \frac{8}{17} \)

Step4: Calculate \( \sin M \)

\( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \), so \( \sin M = \frac{NL}{ML} = \frac{15}{17} \)

Answer:

\( \tan M = \boldsymbol{\frac{15}{8}} \), \( \cos M = \boldsymbol{\frac{8}{17}} \), \( \sin M = \boldsymbol{\frac{15}{17}} \)