Question

try this #1: find side a.

find side a: \frac{a}{\text{sine } a} = \frac{b}{\text{sine } b} = \frac{c}{\text{sine } c}

Explanation:

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

Step 1: Identify the given information

We are given a non-right triangle with the following values:

• Angle \( A = 30^\circ \)
• Angle \( B = 95^\circ \)
• Side \( c = 10\text{ ft} \) (the side opposite to angle \( C \))

We need to find the length of side \( a \) (the side opposite to angle \( A \)).

Step 2: Find the missing angle C

The sum of all interior angles in any triangle is always \( 180^\circ \). We need to find angle \( C \) because we are given the length of side \( c \), and we need a complete side-angle pair to use the Law of Sines.

$$ C = 180^\circ - (A + B) $$

$$ C = 180^\circ - (30^\circ + 95^\circ) $$

$$ C = 180^\circ - 125^\circ $$

$$ C = 55^\circ $$

Step 3: Set up the Law of Sines

The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides:

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

Substitute the known values into the equation:

$$ \frac{a}{\sin 30^\circ} = \frac{10}{\sin 55^\circ} $$

Step 4: Solve for side a

Isolate \( a \) by multiplying both sides of the equation by \( \sin 30^\circ \):

$$ a = \frac{10 \cdot \sin 30^\circ}{\sin 55^\circ} $$

Using the trigonometric values:

• \( \sin 30^\circ = 0.5 \)
• \( \sin 55^\circ \approx 0.8192 \)

$$ a \approx \frac{10 \cdot 0.5}{0.8192} $$

$$ a \approx \frac{5}{0.8192} $$

$$ a \approx 6.10\text{ ft} $$

Answer:

$$ a \approx 6.10\text{ ft} $$