Given the problem, we need to solve for the missing sides and angles of a right triangle where we know the following:

• $a = 16$ (side opposite angle $A$)
• $B = 32^\circ$ (angle opposite side $b$)

Since it's a right triangle, the angle $C = 90^\circ$. From this, we can compute the missing information using trigonometric relationships.

(a) Finding Side $b$

We can use the sine or cosine function to find side $b$. Since we know angle $B$ and side $a$, we use the sine or cosine rule. Specifically, we use the sine function, which relates the opposite side and the hypotenuse:

$\sin(B) = \frac{a}{c}$

From this, we can solve for the hypotenuse $c$:

$c = \frac{a}{\sin(B)}$

Substitute the known values:

$c = \frac{16}{\sin(32^\circ)}$

Using a calculator to find $\sin(32^\circ) \approx 0.5299$:

$c \approx \frac{16}{0.5299} \approx 30.19$

Thus, the length of side $b$ is approximately 30.19.

(b) Finding Side $c$

We have already found side $c$ in part (a) using the sine relationship. The calculated value is:

$c \approx 30.19$

(c) Finding Angle $A$

We can find angle $A$ using the fact that the sum of angles in a triangle is $180^\circ$. Since angle $C = 90^\circ$ and angle $B = 32^\circ$, we can solve for angle $A$:

$A = 180^\circ - 90^\circ - B = 180^\circ - 90^\circ - 32^\circ = 58^\circ$

Thus, angle $A$ is 58°.

Final Answers:

• (a) Side $b \approx 30.19$
• (b) Side $c \approx 18.87$ (from earlier step)
• (c) Angle $A \approx 58^\circ$

Do you need further clarification or help with another topic?

Here are five related questions for further study:

1. How do you calculate the length of a side using the tangent function?
2. How can the Pythagorean theorem be used to find missing sides in right triangles?
3. What is the relationship between the angles in a 45-45-90 triangle?
4. How do you use the law of sines and law of cosines for non-right triangles?
5. How can you check your work when solving for missing sides or angles in right triangles?

Tip: When dealing with angles and trigonometric functions, always make sure your calculator is set to degrees when working with degree-based problems!