Let's analyze and solve the problem.

We are given a right triangle with:

• Angle $B = 17.9^\circ$
• Side $a = 12.6$ (which is the side opposite angle $A$).

We aim to solve the triangle, meaning we need to find:

1. The hypotenuse ($c$),
2. The other leg ($b$),
3. The remaining angle ($A$).

Step 1: Find angle $A$

Since the triangle is a right triangle, the sum of the angles is $90^\circ + 90^\circ = 180^\circ$. Subtract angle $B$ from $90^\circ$ to find $A$: $A = 90^\circ - B = 90^\circ - 17.9^\circ = 72.1^\circ$

Step 2: Use trigonometric functions to find $c$ (hypotenuse)

From the given angle $B = 17.9^\circ$, we know side $a = 12.6$ is the side opposite $A$. Using the sine function: $\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c}$ Rearrange to solve for $c$: $c = \frac{a}{\sin(A)}$ Substitute values: $c = \frac{12.6}{\sin(72.1^\circ)}$ Using a calculator ($\sin(72.1^\circ) \approx 0.9524$): $c \approx \frac{12.6}{0.9524} \approx 13.23$

Step 3: Use trigonometric functions to find $b$ (the other leg)

Now we can use the tangent function with angle $A$ to find $b$. Recall: $\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b}$ Rearrange to solve for $b$: $b = \frac{a}{\tan(A)}$ Substitute values ($\tan(72.1^\circ) \approx 3.078$): $b = \frac{12.6}{3.078} \approx 4.09$

Final Solution:

1. $A = 72.1^\circ$,
2. $b \approx 4.09$,
3. $c \approx 13.23$.

Let me know if you'd like any additional details or further clarifications!

Related Questions:

1. What is the difference between sine, cosine, and tangent?
2. How do you solve a right triangle given two sides instead of one angle?
3. Can you calculate the area of the given triangle?
4. How can the Pythagorean theorem verify the calculations of $c$?
5. How are trigonometric functions related to the unit circle?

Tip:

Always double-check trigonometric calculations by verifying with another function, such as using cosine or the Pythagorean theorem.