Precalculus Study Guide: Trigonometric Identities, Angle Formulas, and Triangle Applications

Q13. Find the distance from A to C, to the nearest yard, using the measurements shown in the figure.

Background

Topic: Law of Cosines (Triangle Applications)

This question tests your ability to apply the Law of Cosines to solve for an unknown side of a triangle when two sides and the included angle are given.

Key formula:

Law of Cosines:

$c^2 = a^2 + b^2 - 2ab \cos(C)$

Where:

• $a$ and $b$ are the known sides (140 yd and 160 yd)

• $C$ is the included angle (80°)

• $c$ is the unknown side (distance from A to C)

Step-by-Step Guidance

1. Identify the known values: $a = 140$ yd, $b = 160$ yd, $C = 80^ ext{o}$.

2. Write the Law of Cosines formula for the unknown side $c$:

   $c^2 = a^2 + b^2 - 2ab \cos(C)$

3. Substitute the known values into the formula:

   $c^2 = (140)^2 + (160)^2 - 2 \times 140 \times 160 \times \cos(80^ ext{o})$

4. Calculate $\cos(80^ ext{o})$ and multiply it by $2 \times 140 \times 160$.

5. Add $(140)^2$ and $(160)^2$, then subtract the value from the previous step.

Try solving on your own before revealing the answer!

Final Answer: AC ≈ 207 yd

$c^2 = 19600 + 25600 - 2 \times 140 \times 160 \times \cos(80^ ext{o})$

After calculating, the distance from A to C is approximately 207 yards. This uses the Law of Cosines to find the missing side given two sides and the included angle.