Question: Solve the right triangle shown in the figure.
B = 17.2°, b = 33.5
A = □ ° (Round to the nearest tenth as needed.)
a ≈ □ (Round to the nearest hundredth as needed.)
c ≈ □ (Round to the nearest hundredth as needed.)
1. **State the problem:**
We are given a right triangle with angle $B = 17.2^\circ$ and side $b = 33.5$ (adjacent to angle $B$). We need to find angle $A$, side $a$, and hypotenuse $c$.
2. **Recall important rules:**
- The sum of angles in a triangle is $180^\circ$.
- In a right triangle, one angle is $90^\circ$.
- Use trigonometric ratios: $\sin$, $\cos$, and $\tan$.
3. **Find angle $A$:**
Since $A + B + 90^\circ = 180^\circ$, then
$$A = 180^\circ - 90^\circ - B = 90^\circ - 17.2^\circ = 72.8^\circ.$$
4. **Find side $a$ (opposite to angle $B$):**
Use the tangent ratio:
$$\tan(B) = \frac{a}{b} \implies a = b \times \tan(B).$$
Calculate:
$$a = 33.5 \times \tan(17.2^\circ).$$
5. **Calculate $a$:**
$$a \approx 33.5 \times 0.3109 = 10.41.$$
6. **Find hypotenuse $c$:**
Use cosine ratio:
$$\cos(B) = \frac{b}{c} \implies c = \frac{b}{\cos(B)}.$$
Calculate:
$$c = \frac{33.5}{\cos(17.2^\circ)}.$$
7. **Calculate $c$:**
$$c \approx \frac{33.5}{0.955} = 35.08.$$
**Final answers:**
- $A = 72.8^\circ$
- $a \approx 10.41$
- $c \approx 35.08$