The given triangle is a right triangle with:

• $\angle B = 90^\circ$,
• $BC = 21$ (the leg adjacent to $\angle A$),
• $AC = 39$ (the hypotenuse),
• and $AB$ is the other leg.

Let's solve step-by-step:

Step 1: Calculate $AB$ using the Pythagorean Theorem

$AC^2 = AB^2 + BC^2$ $39^2 = AB^2 + 21^2$ $1521 = AB^2 + 441$ $AB^2 = 1080$ $AB = \sqrt{1080} = 6\sqrt{30} \approx 32.86$

Step 2: Calculate $\angle A$ using trigonometry

Using the tangent function: $\tan(A) = \frac{AB}{BC}$ $\tan(A) = \frac{32.86}{21}$ $A = \arctan\left(\frac{32.86}{21}\right) \approx 57.12^\circ$

Step 3: Calculate $\angle C$

Since the sum of angles in a triangle is $180^\circ$: $\angle C = 90^\circ - \angle A$ $\angle C = 90^\circ - 57.12^\circ \approx 32.88^\circ$

Final Answers

• $m\angle A \approx 57.12^\circ$
• $m\angle C \approx 32.88^\circ$
• $AB \approx 32.86$
• $AC = 39$
• $BC = 21$

Would you like me to explain any part in more detail or solve another related problem? 😊

Expanding Ideas

1. How do you use sine or cosine to verify the angles of the triangle?
2. What is the exact value of $\sin(\angle A)$ and $\cos(\angle A)$?
3. How do you calculate the area of this triangle?
4. How would you solve if $BC$ and $AB$ were known, but $AC$ wasn't?
5. Can you find the triangle's perimeter?

Tip:

Always check your trigonometric ratios to verify calculations for right triangles!