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:
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1
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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?

2
#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.

#3.
Use the Law of Sines or Law of Cosines. Find the measure x to the nearest tenth.
#1. #2.
3
#4.

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.
4
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? d. 6.7 mi a. 2.3 mi b. 2.6. mi c. 5.1 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.