The problem requires solving a right triangle by finding all missing sides and angles. The triangle has the following given information:

• Angle $A = 39^\circ$
• Angle $C = 51^\circ$ (since the two angles are complementary in a right triangle)
• Side $AB = 19$ (adjacent side to $A$ and opposite to $C$).

We need to find:

1. Hypotenuse ($AC$).
2. Opposite side to $A$ ($BC$).

Step-by-Step Solution:

1. Find $BC$ (opposite side to $A$): Using the tangent function: $\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB}$ Rearrange to find $BC$: $BC = AB \cdot \tan(A) = 19 \cdot \tan(39^\circ)$ $BC = 19 \cdot 0.8098 \approx 15.386$

2. Find $AC$ (the hypotenuse): Using the cosine function: $\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC}$ Rearrange to find $AC$: $AC = \frac{AB}{\cos(A)} = \frac{19}{\cos(39^\circ)}$ $AC = \frac{19}{0.7771} \approx 24.445$

3. Verify the angles: Since this is a right triangle, the angles add up to $90^\circ$: $A + C = 39^\circ + 51^\circ = 90^\circ$

Final Answer:

• $BC = 15.386$
• $AC = 24.445$
• Angles are correct ($A = 39^\circ$, $C = 51^\circ$).

Let me know if you’d like a more detailed explanation for any step!

Related Questions:

1. How can you verify the sides using the Pythagorean theorem?
2. What are some real-life applications of solving right triangles?
3. How would the solution change if $A$ were $45^\circ$?
4. Why is the tangent function helpful for finding the opposite side?
5. How does the cosine function relate to the hypotenuse?

Tip: Always double-check trigonometric values using a calculator or table for accuracy!