Question

1. for each triangle below, write three ratios for the marked angle. do not solve.

a)
(right triangle with legs 14 and 6, hypotenuse 20, marked angle at vertex a)
b)
(right triangle with legs 14 and 8, hypotenuse 15, marked angle at vertex x)

Explanation:

Part a)

Step 1: Identify sides for angle A

In the right - triangle with right - angle at the vertex between sides 6 and 14, for angle A:

• Opposite side (opp) to angle A: \(opp = 6\)
• Adjacent side (adj) to angle A: \(adj=14\)
• Hypotenuse (hyp) of the triangle: \(hyp = 20\)

Step 2: Write sine ratio

The sine of an angle in a right - triangle is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For angle A, \(\sin A=\frac{6}{20}\)

Step 3: Write cosine ratio

The cosine of an angle in a right - triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For angle A, \(\cos A=\frac{14}{20}\)

Step 4: Write tangent ratio

The tangent of an angle in a right - triangle is defined as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For angle A, \(\tan A=\frac{6}{14}\)

Part b)

Step 1: Identify sides for angle X

In the right - triangle with right - angle at the vertex between sides 8 and 14, for angle X:

• Opposite side (opp) to angle X: \(opp = 8\)
• Adjacent side (adj) to angle X: \(adj = 14\)
• Hypotenuse (hyp) of the triangle: \(hyp=15\) (Wait, let's check the Pythagorean theorem: \(8^{2}+14^{2}=64 + 196=260\), but \(15^{2}=225\). There is a mistake here. Wait, maybe the hypotenuse is calculated as \(\sqrt{8^{2}+14^{2}}=\sqrt{64 + 196}=\sqrt{260}=2\sqrt{65}\approx16.12\). But according to the given hypotenuse as 15, maybe it's a typo. But following the given lengths:
• Opposite side (opp) to angle X: \(opp = 8\)
• Adjacent side (adj) to angle X: \(adj = 14\)
• Hypotenuse (hyp) of the triangle: \(hyp = 15\) (assuming the given length is correct for the purpose of ratio writing)

Step 2: Write sine ratio

Using \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), for angle X, \(\sin X=\frac{8}{15}\)

Step 3: Write cosine ratio

Using \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), for angle X, \(\cos X=\frac{14}{15}\)

Step 4: Write tangent ratio

Using \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), for angle X, \(\tan X=\frac{8}{14}\)

Answer:

Part a)

\(\sin A=\frac{6}{20}\), \(\cos A=\frac{14}{20}\), \(\tan A=\frac{6}{14}\)

Part b)

\(\sin X=\frac{8}{15}\), \(\cos X=\frac{14}{15}\), \(\tan X=\frac{8}{14}\) (Note: There is a discrepancy in the hypotenuse length as per Pythagorean theorem, but the ratios are written based on the given side lengths)