Question

which equation can be solved to find one of the missing side lengths in the triangle?

Explanation:

🆕 New Concept Discovered: Right Triangle Trigonometry
Connecting angles and sides in right triangles.

Step 1: Identify the given values

We are given a right-angled triangle \( \triangle ABC \) with:

• The right angle at vertex \( C \) (so the hypotenuse is the side opposite to \( C \), which is \( AB \)).
• Hypotenuse \( AB = 12 \) units.
• Angle \( B = 60^\circ \).
• Adjacent side to angle \( B \) is \( BC = a \).
• Opposite side to angle \( B \) is \( AC = b \).

Step 2: Set up trigonometric ratios

To find the missing side lengths, we use the basic trigonometric definitions (SOH CAH TOA):

1. For the opposite side \( b \):

Using the sine function, which relates the opposite side to the hypotenuse:
\[ \sin(60^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{b}{12} \]

1. For the adjacent side \( a \):

Using the cosine function, which relates the adjacent side to the hypotenuse:
\[ \cos(60^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{a}{12} \]

Answer:

Either of the following equations can be solved to find one of the missing side lengths:

• To find side \( b \):

\[ \sin(60^\circ) = \frac{b}{12} \]

• To find side \( a \):

\[ \cos(60^\circ) = \frac{a}{12} \]