Lesson 7-7
Law of Cosines
Transparency 7-7
5-Minute Check on Lesson 7-6
Find each measure given the measures of ∆RST. Round all side
measurements to the nearest tenth and angles to the nearest degree..
1. Find s, if mR = 63°, mS = 38°, and r = 52.
35.9
2. Find mR, if mS = 122°, s = 10.8, and r = 5.2.
24°
Solve ∆MNP described below. Round all side measurements to the
nearest tenth and angles to the nearest degree.
3. mM = 50°, if mN = 32°, and m = 15. mP = 98°, n = 10.4, p = 19.4
4. n = 8.5, p = 10.8, and mP = 110°. mN = 48°, mM = 22°, m = 4.4
5. Standardized Test Practice: Find the perimeter of
quadrilateral ABCD to the nearest tenth.
70°
8 cm
54°
A
27.6
B
29.8
C
32.0
D
Click the mouse button or press the
Space Bar to display the answers.
34.6
Objectives
• Use the Law of Cosines to solve triangles
• Solve problems by using the Law of Cosines
Vocabulary
• None new
Law of Cosines
A
Let ∆ABC be any triangle with a, b and c
representing the measures of the sides
opposite the angles with measures A, B,
and C respectively. Then the following
equations are true:
a2 = b2 + c2 – 2bc cos A
b
c
B
a
b2 = a2 + c2 – 2ac cos B
c2 = a2 + b2 – 2ab cos C
Law of Cosines can be used to solve triangles when the Law of Sines
cannot be used
Case 1: measures of two sides and their included angle (SAS)
Case 2: measures of all three sides (SSS)
C
Example1
Use the Law of Cosines
since the measures of two
sides and the included angle
are known.
Law of Cosines
Simplify.
Take the square root
of each side.
Use a calculator.
Answer:
Example 2
Answer:
Example 3
Law of Cosines
Simplify.
Subtract 754 from each side.
Divide each side by –270.
Solve for L.
Use a calculator.
Answer:
Example 4
Answer:
Example 5
Determine whether the Law of Sines or the Law of
Cosines should be used first to solve
Then
solve
Round angle measures to the nearest
degree and side measures to the nearest tenth.
Since we know the measures of
two sides and the included
angle, use the Law of Cosines.
Law of Cosines
Take the square root
of each side.
Use a calculator.
Example 5 cont
Next, we can find
If we decide to find
we can use either the Law of Sines or the Law of Cosines
to find this value. In this case, we will use the Law of
Sines.
Law of Sines
Cross products
Divide each side by 46.9.
Example 5 cont
Take the inverse of each
side.
Use a calculator.
Use the Angle Sum Theorem to find
Angle Sum Theorem
Subtract 168 from each
side.
Answer:
Example 6
Determine whether the Law of Sines or the Law of
Cosines should be used first to solve
Then
solve
Round angle measures to the nearest
degree and side measures to the nearest tenth.
Answer:
Summary & Homework
• Summary:
– The Law of Cosines can be used to solve non-right
triangles when you know
1) the measures of two sides and the included
angle (SAS) or
2) the measures of all three sides (SSS)
• Homework:
– pg 388-389; 11, 12, 19-21, 27-30, 42