Question

1. use the given information about $\triangle abc$ with right angle $b$ to find the unknown side lengths. round your answer to the nearest hundredth. $m\angle c = 75^{\circ}$ and $b = 24$

a $c \approx 23.18, a \approx 6.21$
b $c \approx 25.17, a \approx 8.47$
c $c \approx 11.24, a \approx 3.27$
d $c \approx 17.48, a \approx 7.15$

Explanation:

Step1: Identify sides in right triangle

In right triangle \( \triangle ABC \) with right angle at \( B \), \( b \) is the hypotenuse (since it's opposite the right angle), \( \angle C = 75^\circ \), so side \( c \) is opposite \( \angle C \), and side \( a \) is opposite the remaining angle ( \( 15^\circ \), since \( 90 - 75 = 15 \)).

Step2: Use sine and cosine ratios

• Sine of \( \angle C \): \( \sin(75^\circ) = \frac{c}{b} \), so \( c = b \cdot \sin(75^\circ) \).
• Cosine of \( \angle C \): \( \cos(75^\circ) = \frac{a}{b} \), so \( a = b \cdot \cos(75^\circ) \).

Step3: Calculate \( c \) and \( a \)

• \( \sin(75^\circ) \approx 0.9659 \), \( b = 24 \), so \( c = 24 \times 0.9659 \approx 23.18 \).
• \( \cos(75^\circ) \approx 0.2588 \), so \( a = 24 \times 0.2588 \approx 6.21 \).

Answer:

a. \( c \approx 23.18, a \approx 6.21 \)