Download Lesson #3: Cosine Law

Lesson #3: Cosine Law
The Sine Law allows you to solve for triangles where:
• 2 sides and a corresponding angle are known (SSA or side-side-angle)
• 2 angles and a corresponding side (AAS or angle-angle-side)
The Cosine Law allows you to solve for:
• The 3rd side of the triangle if you know 2 sides of a triangle and the angle
that is formed between these two sides. (SAS or side – angle – side)
• An angle if you know the three side lengths of the triangle.
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The Cosine Law formula:
To Find A Side:
a2 = b2 + c2 – 2bc cos A
To Find An Angle:
cos A = b2 + c2 – a2
(2bc)
Now that you know another trig formula, remember…
 When solving triangles:

Check for right angles (900).  Use basic trigonometric ratio’s (SOH CAH TOA)

Check for Sine Law Ratios (a side and an opposite angle).  Use Sine Law.

If none of the above possibilities exist:  Use Cosine Law.
Example 1: Write the cosine formula for the missing side R of the following
triangle PQR.
To Find A Side:
a2 = b2 + c2 – 2bc cos A
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Example 2: Write the cosine formula for the missing angle Q of the following
triangle PQR.
To Find An Angle:
cos A = b2 + c2 – a2
(2bc)
Example 3: Find the measure of angle Y.
X
z = 15
Y
Note: It’s a
SSS
triangle
y = 17
x = 20
To Find An Angle:
Z
cos A = b2 + c2 – a2
(2bc)
Find angle Y
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Example 4: Find the measure of side a, angle B, and angle C.
First check…
1. Right angle triangle?
2. Sine-Law ratio’s? (opposites?)
3. If no to both, use Cosine Law
A
55°
18 cm
Note: It’s a
SAS
triangle
14 cm
C
B
To Find A Side:
a2 = b2 + c2 – 2bc cos A
First: Find side a
Second: Find angle B
Third: Find angle C
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Assignment #3: Cosine Law
1. Given ∆ABC. Solve for side a.
A
c = 350
68°
b= 475
B
C
2. Given ∆ABC. Solve for ∠A.
A
c = 55
b = 75
B
a = 70
C
3. From a lighthouse, a cruise ship can be seen 8.3 km away and a freighter can also be
seen 12.5 km away. How far away is the cruise ship from the freighter if the angle
between the lines of observation are 68°?
Lighthouse
8.3
Cruise Ship
68°
12.5
?
Freighter
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4. Solve for all the interior angles.
A
c = 18 cm
B
b = 20 cm
a = 19 cm
C
NOTE: For the following questions, make sure to draw a diagram to help you with
the question.
5. At a provincial park, there is a sign, a reception area, and a picnic area. The
reception area is 350 m away from the picnic area, the picnic area is 475 m away
from the sign. From the picnic area, the angle between the 2 lines of sight for the
reception area and the sign is 64°. How far apart is the sign from the reception
area?
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6.
An Art Gallery is in the shape of a triangle. Two of the walls are 114 m and 61 m in
length. The angle between these 2 walls is 72°.
a. How long is the 3rd wall?
b. What are the angles of the other 2 corners of the triangle?
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7. Construction has been started on a building as shown by the diagram.
12 ft
Pier
10 ft
Braces
a. What is the length of each brace?
b. What is the angle between both braces?
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