Question

1. find the measure of angle a.

• 20.67^
• 21.75^
• 22.50^
• 19.50^

Explanation:

🆕 New Concept Discovered: Law of Sines
Using ratios to find missing angles in non-right triangles.

Step 1: Identify the given information

We are given a triangle \(ABC\) with the following known values:

• Side \(b\) (opposite to angle \(B\)) = \(43\)
• Side \(a\) (opposite to angle \(A\)) = \(19\)
• Angle \(B\) = \(123^\circ\)

We need to find the measure of angle \(A\).

Step 2: Set up the Law of Sines

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles:

$$ \frac{\sin(A)}{a} = \frac{\sin(B)}{b} $$

Substitute the known values into the formula:

$$ \frac{\sin(A)}{19} = \frac{\sin(123^\circ)}{43} $$

Step 3: Solve for \(\sin(A)\)

Isolate \(\sin(A)\) by multiplying both sides by \(19\):

$$ \sin(A) = \frac{19 \cdot \sin(123^\circ)}{43} $$

Using a calculator to find the value of \(\sin(123^\circ) \approx 0.83867\):

$$ \sin(A) \approx \frac{19 \cdot 0.83867}{43} $$

$$ \sin(A) \approx \frac{15.9347}{43} $$

$$ \sin(A) \approx 0.37057 $$

Step 4: Calculate the angle \(A\)

Find the inverse sine (\(\arcsin\)) of \(0.37057\) to find the measure of angle \(A\):

$$ A = \arcsin(0.37057) $$

$$ A \approx 21.75^\circ $$

Answer:

\(21.75^\circ\)