We are solving a right triangle problem. The provided information includes:

• Angle $B = 30^\circ$,
• Hypotenuse $c = 15.8 \, \text{units}$.

We need to calculate the following:

1. The length of side $b$ (the base),
2. The length of side $a$ (the height),
3. The measure of angle $A$.

Step 1: Recall basic trigonometric relations

In a right triangle:

• $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$,
• $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$,
• $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Here:

• $\sin(30^\circ) = 0.5$,
• $\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$.

Step 2: Solve for $a$ (opposite side)

Using $\sin(B) = \frac{a}{c}$: $\sin(30^\circ) = \frac{a}{15.8}.$ Substitute $\sin(30^\circ) = 0.5$: $0.5 = \frac{a}{15.8}.$ Solve for $a$: $a = 15.8 \cdot 0.5 = 7.9 \, \text{units}.$

Step 3: Solve for $b$ (adjacent side)

Using $\cos(B) = \frac{b}{c}$: $\cos(30^\circ) = \frac{b}{15.8}.$ Substitute $\cos(30^\circ) = 0.866$: $0.866 = \frac{b}{15.8}.$ Solve for $b$: $b = 15.8 \cdot 0.866 \approx 13.69 \, \text{units}.$

Step 4: Solve for angle $A$

In a right triangle, the sum of the angles is $180^\circ$, and one angle is $90^\circ$. So: $A = 90^\circ - B = 90^\circ - 30^\circ = 60^\circ.$

Final Results

1. $b = 13.69 \, \text{units}$,
2. $a = 7.9 \, \text{units}$,
3. $A = 60^\circ$.

Would you like me to go into more detail about any of the steps?

Relative Questions:

1. How can trigonometric functions be applied to other right triangle problems?
2. What is the significance of the 30-60-90 triangle rule in solving this problem?
3. Can you derive exact trigonometric ratios for $30^\circ$, $60^\circ$, and $45^\circ$?
4. What are the practical applications of such triangle calculations?
5. How can we verify the solution using the Pythagorean theorem?

Tip:

Memorize the special right triangle ratios for $30^\circ$-$60^\circ$-$90^\circ$ triangles: $1:\sqrt{3}:2$ (shorter leg, longer leg, hypotenuse). This helps solve these problems faster!