COSINE LAW (SIDES)
When do you use cosine law
When you know 2 sides and the included angle OR all 3 sides
Explain when you solve for an angle vs a side in Cosine Law.
Side: use full formula a² = b² + c² − 2bc cos(A)
Angle: rearrange to cos(A) = (b² + c² − a²) / 2bc
Find GCF of 6 and 9
3
Factor x² + 4x + 3
: (x + 1)(x + 3)
7⁰
1
Write the Cosine lw for finding a side
\(a\):
\(a^2 = b^2 + c^2 - 2bc \cos(A)\) [1]
a = 5, b = 6, c = 7. Find angle A.
a = 5, b = 6, c = 7
cos A = (6^2 + 7^2 - 5^2) / (267)
cos A = (36 + 49 - 25) / 84
cos A = 60 / 84 = 0.7143
A = cos-1(0.7143) ≈ 44°
Find GCF of 12x and 18x²
6x
Factor x² + 6x + 8
(x + 2)(x + 4)
2⁻²
1/4
A triangle has sides b = 6, c = 8, and angle A = 60°. Find side a
Given: b = 6, c = 8, A = 60°
Formula: a² = b² + c² − 2bc cos(A)
Substitute:
a² = 6² + 8² − 2(6)(8)cos(60°)
Solve:
a ≈ 7
a = 8, b = 10, c = 12 find the missing angle
cos A = (10^2 + 12^2 - 8^2) / (21012)
cos A = (100 + 144 - 64) / 240
cos A = 180 / 240 = 0.75
A = cos-1(0.75) ≈ 41°
Find GCF of 20a² and 30a³
10a²
x² + 9x + 20
(x + 4)(x + 5)
x³ · x⁵
x⁸
In a triangle,
b = 9, c = 12, A = 45°.
Find side a.
Formula:
a² = b² + c² − 2bc cos A
a² = 9² + 12² − 2(9)(12)cos45°
a² = 81 + 144 − 216(0.7071)
a² = 225 − 152.36
a² ≈ 72.64
a ≈ √72.64
A: a ≈ 8.5
a = 13, b = 9, c = 11 find the missing angle
cos A = (9^2 + 11^2 - 13^2) / (2911)
cos A = (81 + 121 - 169) / 198
cos A = 33 / 198 = 0.1667
A = cos-1(0.1667) ≈ 80
Factor: 14x + 21
7(2x + 3)
Factor x² + 11x + 24
(x + 3)(x + 8)
(x²)³
x⁶
In a triangle,
b = 14, c = 10, A = 120°.
Find side a.
Formula:
a² = b² + c² − 2bc cos A
a² = 14² + 10² − 2(14)(10)cos120°
a² = 196 + 100 − 280(−0.5)
a² = 296 + 140
a² = 436
a ≈ √436
a ≈ 20.9
A: a ≈ 20.9
a = 20, b = 13, c = 17 find the missing angle
cos A = (13^2 + 17^2 - 20^2) / (21317)
cos A = (169 + 289 - 400) / 442
cos A = 58 / 442 = 0.1312
A = cos-1(0.1312) ≈ 83°
Factor: 24x² − 36x
12x(2x − 3)
Factor x² + 14x + 45
(x + 5)(x + 9)
(3x²)²
9x⁴