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Algebra II – Chapter 12 Day #4
Topic: The Law of Cosines & Multi-Step Trig Problems
Standards/Goals:
 G.SRT.11.: I can find the sides and angles of a triangle using the Law of Cosines.
 G.3.a.: I can use the law of cosines to find the lengths and measures of angles in triangles in
mathematical and real-world problems.
Today, we are going to learn another way to find angles and sides of triangles, when a right angle is
not inside the triangle. This new method is known as the Law of Cosines.
Law of Cosines:
In ΔABC, let a, b, and c represent the lengths of the sides opposite <A, <B and <C, respectively:
𝒂𝟐 = 𝒃𝟐 + 𝒄𝟐 − 𝟐𝒃𝒄 𝒄𝒐𝒔𝑨
𝒃𝟐 = 𝒂𝟐 + 𝒄𝟐 − 𝟐𝒂𝒄 𝒄𝒐𝒔𝑩
𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 − 𝟐𝒂𝒃 𝒄𝒐𝒔𝑪
Let’s try some example problems that will utilize the Law of Cosines.
EXAMPLES:
#1. What is the measure of <C? Round your answer to the nearest tenth of a degree.
#2. In ΔABC, b = 8.4, c = 9.6 and m<A = 30 degrees. What is m<B?
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#3. What is the measure of <C? Round your answer to
the nearest TENTH of a degree.
#4. WORD PROBLEM: A map shows the location of three cities that form a triangle with a 72 degree
angle between two sides that represent distances 2.8 mi and 4.2 mi. What is the distance represented
by the length of the third side?
We finally, want to be able to work problems that require multiple steps using trigonometry.
EXAMPLES: Find the length of the side labeled ‘x.’ Round intermediate values to the nearest tenth. Use
the rounded values to calculate the next value. Round your answer to the nearest tenth.
#1.
#2.
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HOMEWORK – Chapter 12 Day #4
Name ______________________________________ Date _______
Use the Law of Sines or Law of Cosines. Find the measure x to the nearest tenth.
#1.
#2.
#3.
#4.
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PROBLEM SOLVING:
Find the area of each triangle. Round intermediate values to the nearest tenth. Use the
rounded values to calculate the next value. Round your final answer to the nearest tenth.
Use the Law of Cosines to find each value to the nearest tenth.
#7. Find a if A = 18°, c = 72, and b = 100.
#8. Find A if B = 45°, a = 9, and c = 19.
#9. MULTIPLE CHOICE: The shape of a piece of property is triangular with an 85 degre angle
between two sides of 1.2 mi and 2.4 mi. What is the approximate length of the third side?
a. 2.3 mi
b. 2.6. mi
c. 5.1 mi
d. 6.7 mi
#10. Find the length of the side labeled ‘x.’ Round
intermediate values to the nearest tenth. Use the
rounded values to calculate the next value. Round
your answer to the nearest tenth.