Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 7 - Applications of Trigonometry and Vectors - Section 7.1 Oblique Triangles and the Law of Sines - 7.1 Exercises: 25

Answer

$AB=118m$

Work Step by Step

$$B=112^\circ10',C=15^\circ20',BC=a=354m$$ $$AB=c=?$$ To find $c$, we look at the given information. $a$ is already known, so we can apply the following law of sines: $$\frac{c}{\sin C}=\frac{a}{\sin A}$$ $$c=\frac{a\sin C}{\sin A}$$ In this equation, $a$ and $C$ are known, but $A$ is unknown. However, we already know $B$ and $C$, so it is easy to calculate $A$ right away from the fact that the sum of 3 angles in any triangle equals $180^\circ$. 1) Find $A$ As the sum of 3 angles in any triangle equals $180^\circ$: $$A+B+C=180^\circ$$ $$A+112^\circ10'+15^\circ20'=180^\circ$$ $$A+127^\circ30'=180^\circ$$ $$A=52^\circ30'=52.5^\circ$$ For calculation, we also need to change angle $C$ to complete degree: $$C=15^\circ20'\approx15.333^\circ$$ 2) Find $c$ $$c=\frac{a\sin C}{\sin A}$$ $$c=\frac{354\sin15.333^\circ}{\sin52.5^\circ}$$ $$c\approx118m$$ That means $AB=118m$
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