Question: ∠B = [ ]
Round your answer to the nearest hundredth.
C
5
A
2
?
B
A right triangle is shown in the center of the page with a right angle at C, side CA labeled 5, side CB labeled 2, and angle B marked with a question mark; position_hint \in {center}
1. **State the problem:** We need to find the measure of angle $B$ in a right triangle $ABC$ where the right angle is at $C$, side $CA = 5$, and side $CB = 2$.
2. **Identify the sides relative to angle $B$:**
- Side $CB$ is adjacent to angle $B$.
- Side $CA$ is opposite to angle $B$.
- Side $AB$ is the hypotenuse.
3. **Use the Pythagorean theorem to find the hypotenuse $AB$:**
$$AB = \sqrt{CA^2 + CB^2} = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29}$$
4. **Use the tangent function to find angle $B$:**
$$\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{CA}{CB} = \frac{5}{2}$$
5. **Calculate angle $B$ using the arctangent function:**
$$B = \tan^{-1}\left(\frac{5}{2}\right)$$
6. **Evaluate the arctangent:**
$$B \approx \tan^{-1}(2.5) \approx 68.20^\circ$$
7. **Final answer:**
$$\boxed{68.20^\circ}$$
This is the measure of angle $B$ rounded to the nearest hundredth.