Question

1. find the three trigonometric ratios of each of the acute angles in the triangle.

a
$sin n = \frac{5}{4}, \cos n = \frac{3}{4}, \tan n = \frac{5}{3};$
$sin m = \frac{3}{4}, \cos m = \frac{5}{4}, \tan m = \frac{5}{3}$
b
$sin n = \frac{4}{5}, \cos n = \frac{3}{5}, \tan n = \frac{4}{3};$
$sin m = \frac{3}{5}, \cos m = \frac{4}{5}, \tan m = \frac{3}{4}$
c
$sin n = \frac{3}{5}, \cos n = \frac{4}{5}, \tan n = \frac{3}{4};$
$sin m = \frac{4}{5}, \cos m = \frac{3}{5}, \tan m = \frac{4}{3}$

Explanation:

Step1: Recall Trigonometric Ratios

For a right triangle, the trigonometric ratios are:

• Sine (sin) of an angle = $\frac{\text{opposite side}}{\text{hypotenuse}}$
• Cosine (cos) of an angle = $\frac{\text{adjacent side}}{\text{hypotenuse}}$
• Tangent (tan) of an angle = $\frac{\text{opposite side}}{\text{adjacent side}}$

The hypotenuse is the side opposite the right angle (here, $NM = 15$). The legs are $ON = 12$ (opposite to $\angle M$) and $OM = 9$ (opposite to $\angle N$).

Step2: Analyze $\angle N$

• Opposite side to $\angle N$: $OM = 9$
• Adjacent side to $\angle N$: $ON = 12$
• Hypotenuse: $NM = 15$

• $\sin N = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{15} = \frac{3}{5}$
• $\cos N = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{15} = \frac{4}{5}$
• $\tan N = \frac{\text{opposite}}{\text{adjacent}} = \frac{9}{12} = \frac{3}{4}$

Step3: Analyze $\angle M$

• Opposite side to $\angle M$: $ON = 12$
• Adjacent side to $\angle M$: $OM = 9$
• Hypotenuse: $NM = 15$

• $\sin M = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{15} = \frac{4}{5}$
• $\cos M = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{9}{15} = \frac{3}{5}$
• $\tan M = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{9} = \frac{4}{3}$

Step4: Match with Options

Comparing with the given options:

• For $\angle N$: $\sin N = \frac{3}{5}$, $\cos N = \frac{4}{5}$, $\tan N = \frac{3}{4}$
• For $\angle M$: $\sin M = \frac{4}{5}$, $\cos M = \frac{3}{5}$, $\tan M = \frac{4}{3}$

This matches option c.

Answer:

c. $\sin N = \frac{3}{5}, \cos N = \frac{4}{5}, \tan N = \frac{3}{4}; \sin M = \frac{4}{5}, \cos M = \frac{3}{5}, \tan M = \frac{4}{3}$