To solve this right triangle with $B = 32.5^\circ$ and $c = 27.8$, we need to find the remaining angle $A$, as well as the lengths $a$ and $b$.

Step-by-Step Solution

1. Calculate Angle $A$: Since it’s a right triangle, we know that: $A + B = 90^\circ$ Therefore: $A = 90^\circ - 32.5^\circ = 57.5^\circ$

2. Calculate Side $a$: Using the sine function: $\sin(B) = \frac{a}{c}$ Rearranging to solve for $a$: $a = c \cdot \sin(B)$ Plugging in $c = 27.8$ and $B = 32.5^\circ$: $a = 27.8 \cdot \sin(32.5^\circ)$ Calculating this gives: $a \approx 27.8 \cdot 0.5373 = 14.94$ (rounded to three significant digits as the angle is given to the nearest 0.1°).

3. Calculate Side $b$: Using the cosine function: $\cos(B) = \frac{b}{c}$ Solving for $b$: $b = c \cdot \cos(B)$ Plugging in the values: $b = 27.8 \cdot \cos(32.5^\circ)$ Calculating this gives: $b \approx 27.8 \cdot 0.8442 = 23.5$ (again rounded to three significant digits).

Final Answer:

• Angle $A$: $57.5^\circ$
• Side $a$: $14.9$
• Side $b$: $23.5$

The correct answer would be option A with:

• $A = 57.5^\circ$
• $a = 14.9$
• $b = 23.5$

Would you like further details or explanations on any part of this solution?

Related Questions to Explore:

1. How would you solve the triangle if you were only given angle $A$ and side $a$?
2. What if the angle $B$ was given with a precision of 0.01° instead of 0.1°?
3. Can we solve this triangle using the Pythagorean theorem instead of trigonometric functions?
4. How does rounding affect the accuracy of our answers in trigonometric calculations?
5. What are some real-life applications of solving right triangles like this?

Quick Tip:

Always ensure that angle measurements and significant digits align with the given accuracy requirements for the best precision in trigonometry problems.