1. **State the problem:** We have a right triangle with angle $A = 32.4^\circ$, side $b = 48$ opposite angle $B$, and right angle $C = 90^\circ$. We need to find angle $B$, side $a$ opposite angle $A$, 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 the Pythagorean theorem: $$c = \sqrt{a^2 + b^2}$$ - Use trigonometric ratios: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ 3. **Find angle $B$:** $$B = 90^\circ - A = 90^\circ - 32.4^\circ = 57.6^\circ$$ 4. **Find side $a$ using tangent:** $$\tan(A) = \frac{a}{b} \implies a = b \times \tan(A)$$ Calculate: $$a = 48 \times \tan(32.4^\circ)$$ Using a calculator: $$a \approx 48 \times 0.6357 = 30.51$$ 5. **Find hypotenuse $c$ using Pythagorean theorem:** $$c = \sqrt{a^2 + b^2} = \sqrt{30.51^2 + 48^2}$$ Calculate: $$c = \sqrt{930.72 + 2304} = \sqrt{3234.72} \approx 56.88$$ **Final answers:** $$B = 57.6^\circ, \quad a = 30.51, \quad c = 56.88$$