Questions: Solve the right triangle with α=39° 17′ a=9.38. b ≈ (Do not round until the final answer. Then round to the nearest hundredth as needed.) c ≈ (Do not round until the final answer. Then round to the nearest hundredth as needed.) β ≈ ° (Do not round until the final answer. Then round to the nearest whole number as needed.)

Solve the right triangle with α=39° 17′ a=9.38.
b ≈
(Do not round until the final answer. Then round to the nearest hundredth as needed.)
c ≈
(Do not round until the final answer. Then round to the nearest hundredth as needed.)
β ≈ °
(Do not round until the final answer. Then round to the nearest whole number as needed.)

Transcript text: Solve the right triangle with $\alpha=39^{\circ} 17^{\prime} \mathrm{a}=9.38$. \[ b \approx \] (Do not round until the final answer. Then round to the nearest hundredth as needed.) $c \approx$ (Do not round until the final answer. Then round to the nearest hundredth as needed.) \[ \beta \approx \square^{\circ} \square \] (Do not round until the final answer. Then round to the nearest whole number as needed.)

Solution

Solution Steps

To solve the right triangle, we need to find the missing side lengths $ b $ and $ c $, and the angle $ \beta $. Given $ \alpha = 39^\circ 17' $ and $ a = 9.38 $:

Convert the angle $ \alpha $ from degrees and minutes to decimal degrees.
Use the trigonometric identity $\sin(\alpha) = \frac{a}{c}$ to find $ c $.
Use the trigonometric identity $\tan(\alpha) = \frac{a}{b}$ to find $ b $.
Calculate $ \beta $ using the fact that the sum of angles in a triangle is $ 180^\circ $, and one angle is $ 90^\circ $.

Step 1: Convert Angle to Decimal Degrees

The given angle $\alpha$ is $39^\circ 17'$. To convert this to decimal degrees, we use the formula: \[ \alpha = 39 + \frac{17}{60} = 39.2833^\circ \]

Step 2: Calculate Hypotenuse $c$

Using the trigonometric identity $\sin(\alpha) = \frac{a}{c}$, we solve for $c$: \[ c = \frac{a}{\sin(\alpha)} = \frac{9.38}{\sin(39.2833^\circ)} \approx 14.8147 \]

Step 3: Calculate Side $b$

Using the trigonometric identity $\tan(\alpha) = \frac{a}{b}$, we solve for $b$: \[ b = \frac{a}{\tan(\alpha)} = \frac{9.38}{\tan(39.2833^\circ)} \approx 11.4669 \]

Step 4: Calculate Angle $\beta$

Since the sum of angles in a triangle is $180^\circ$ and one angle is $90^\circ$, we find $\beta$ as follows: \[ \beta = 90^\circ - \alpha = 90^\circ - 39.2833^\circ \approx 50.7167^\circ \]