AnswersPrecalculusLaw of Sines and Law of Cosines — a Deeper Look

Law of Sines and Law of Cosines — a Deeper Look Answers

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To start the proof of the Law of Cosines, an altitude is constructed. Which of the following is formed due to the construction of the altitude in a non-right triangle?

A
one right triangle
B
one acute triangle
C
two right triangles
D
two acute triangles
2
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Which equation below is a true statement about ?

Question illustration
A
x = b minus a cosine C
Option A
B
cosine C = StartFraction h over a EndFraction
Option B
C
h = a cosine C
Option C
D
cosine C = StartFraction h over b EndFraction
Option D
3

Which equation could be used to solve for the length of side c, given a = 5, b = 12, and C = 72°? [Not drawn to scale]

Question illustration
A
square of c = square of 5 + square of 12 minus 2(5)(12) cosine 72 degree
Option A
B
square of c = square of 5 + square of 12 + 2(5)(12) cosine 72 degree
Option B
C
square of c = square of 5 minus square of 12 minus 2(5)(12) cosine 72 degree
Option C
D
square of c = square of 5 minus square of 12 + 2(5)(12) cosine 72 degree
Option D
4

Which equation is a true statement about below?

Question illustration
A
square of h = square of a + square of b
Option A
B
square of h = square of a minus square of b
Option B
C
square of h = square of a + square of x
Option C
D
square of h = square of a minus square of x
Option D
5

Kyung is building a triangular pool in his backyard. He knows that side f is 12 m and that both angles G and F are 40°, which means that angle D is 100°. What is the length of side d? Round the answer to the nearest tenth. [Not drawn to scale]

Question illustration
A
7.6 m
B
7.8 m
C
18.4 m
D
19.0 m
6

Which equation below is a true statement about ?

Question illustration
A
sine A = StartFraction h over b EndFraction
Option A
B
sine C = StartFraction a over h EndFraction
Option B
C
sine A = StartFraction h over c EndFraction
Option C
D
sine C = StartFraction h over b EndFraction
Option D
7

In the proof of the Law of Cosines, the equation was created using the Pythagorean theorem. Which equation is a result of expanding ?

Question illustration
A
square of c = square of h + square of b minus square of x
Option A
B
square of c = square of h + square of b minus 2bx + square of x
Option B
C
square of c = square of h + square of b + square of x
Option C
D
square of c = square of h + square of b minus 2bx minus square of x
Option D
8

What is the length of side c, given a = 10, b = 8, and C = 105°? Round the answer to the nearest tenth. [Not drawn to scale]

Question illustration
A
2.3
B
8.8
C
11.1
D
14.3
9

Eddie solved for the length of side c in using the Law of Cosines formula, . Sophia wants to solve for the side length of b. Which equation could Sophia use to solve for the side length of b, given the other two side lengths and the measure of their included angle?

Question illustration
A
Square of b = Square of a + Square of c minus 2ac cosine C
Option A
B
Square of b = Square of a + Square of c minus 2ab cosine C
Option B
C
Square of b = Square of a + Square of c minus 2ab cosine A
Option C
D
Square of b = Square of a + Square of c minus 2ac cosine B
Option D
10

What is the measure of angle R in shown below, given r = 10, q = 20, and Q = 100°? Round the answer to the nearest tenth. [Not drawn to scale]

Question illustration
A
0.5°
B
2.0°
C
29.5°
D
50.5°

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