Question

try this #2: find angle a, side b, and side c.

Explanation:

🆕 New Concept Discovered: Law of Sines
Relating sides and angles in non-right triangles

Step 1: Find Angle A

The sum of all interior angles in any triangle is always \( 180^\circ \). We can find the measure of angle \( A \) by subtracting the two known angles from \( 180^\circ \):

$$ A = 180^\circ - B - C $$

$$ A = 180^\circ - 141^\circ - 23^\circ $$

$$ A = 16^\circ $$

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Step 2: Find Side b

To find the missing sides of a non-right triangle when we know its angles and at least one side, we use the Law of Sines. This law states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides:

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

From the given triangle:

• The side opposite to angle \( A \) is side \( a = 9 \).
• The side opposite to angle \( B \) is side \( b \).
• The side opposite to angle \( C \) is side \( c \).

Using the ratio of the known side \( a \) and angle \( A \), we set up the equation to solve for \( b \):

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

$$ \frac{9}{\sin(16^\circ)} = \frac{b}{\sin(141^\circ)} $$

Solve for \( b \):

$$ b = \frac{9 \cdot \sin(141^\circ)}{\sin(16^\circ)} $$

Using a calculator to find the sine values:

• \( \sin(141^\circ) \approx 0.6293 \)
• \( \sin(16^\circ) \approx 0.2756 \)

$$ b \approx \frac{9 \cdot 0.6293}{0.2756} \approx 20.55 $$

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Step 3: Find Side c

Similarly, we use the Law of Sines to set up the equation for side \( c \):

$$ \frac{a}{\sin(A)} = \frac{c}{\sin(C)} $$

$$ \frac{9}{\sin(16^\circ)} = \frac{c}{\sin(23^\circ)} $$

Solve for \( c \):

$$ c = \frac{9 \cdot \sin(23^\circ)}{\sin(16^\circ)} $$

Using a calculator to find the sine value:

• \( \sin(23^\circ) \approx 0.3907 \)

$$ c \approx \frac{9 \cdot 0.3907}{0.2756} \approx 12.76 $$

Answer:

• Angle A = \( 16^\circ \)
• Side b \approx \( 20.55 \)
• Side c \approx \( 12.76 \)