Download Lesson 5.4 The Law of Cosines notes

5.4 The Law of Cosines
It is possible to solve an oblique (nonright) triangle whenever the measures of the three sides or two
sides and the included angle are given by using the Law of Cosines.
A
C
c
b
a
b
C
B
a
B
A
c
Law of Cosines
For any ABC, where a, b, and c are the lengths of the sides
opposite the angles with measures A, B, and C, respectively,
a2 = b2 + c2 – 2bc cos A
or
cos 𝐴 =
b2 = a2 + c2 – 2ac cos B
or
cos 𝐵 =
c2 = a2 + b2 – 2ab cos C
or
cos 𝐶 =
𝑏2 +𝑐 2 −𝑎2
2𝑏𝑐
𝑎2 +𝑐 2 −𝑏2
2𝑎𝑐
𝑎2 +𝑏2 −𝑐 2
2𝑎𝑏
Given Information
Appropriate Law
Three sides
Law of Cosines
Two sides and the included angle
Law of Cosines
Two sides and an angle opposite one side
(ambiguous case)
Law of Sines
One side and two angles
Law of Sines
Advanced Mathematics/Trigonometry: Lesson 5.4 The Law of Cosines
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Example 1: Solve each of the following triangles. Express answers to the nearest tenth of a unit.
a. ABC: mC = 100.5, a = 1.2, and b = 2.6
b. ABC: mC = 96.4, a = 5.8, and b = 8.3
c. RST: mS = 75.8, r = 51.9, and t = 36.2
Advanced Mathematics/Trigonometry: Lesson 5.4 The Law of Cosines
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Example 2: Solve each of the following triangles. Express angle measures to the nearest degree.
a. ABC: a = 14.3, b = 10.6, and c = 8.4
b. PQR: p = 17.2, q = 15.4, and r = 13.6
c. RST: r = 27.8, s = 31.3, and t = 23.5
Advanced Mathematics/Trigonometry: Lesson 5.4 The Law of Cosines
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Example 3: Use the law of cosines to find each solution.
a. Two airplanes leave an airport at the same time. The heading of the first is 120 and the heading
of the second is 320. If the planes travel at rates of 700 and 600 mi/h, respectively, how far
apart are they after 1 h?
b. Two airplanes leave an airport at the same time. The heading of the first is 150 and the heading
of the second is 260. If the planes travel at rates of 680 and 560 mi/h, respectively, how far
apart are they after 2 h?
Homework: p. 268 => Class Exercises 1 - 8
pp. 269 – 270 => Practice Exercises 1 – 10; 14; 17; 18; 21; 22; 28; 29; 32
Advanced Mathematics/Trigonometry: Lesson 5.4 The Law of Cosines
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