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Law of cosines or the cosine law helps find out the value of unknown angles or sides on a triangle. This law uses the rules of the Pythagorean theorem. The pythagorean theorem works for right-angled triangles, while this law works for other triangles without a right angle. This law can be used to find the length of one side of a triangle when the lengths of the other 2 sides are given, and the opposite angle of the unknown side is known. In order to find a specific angle using this law, the values of the lengths of all the sides need to be known.
Key Takeaways: Cosine Law, Pythagorean Theorem, Sine Law, Angles, Triangles
What is Cosine Law?
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Cosine law, like the sine law, is a rule used to find an unknown side or angle on a triangle. The cosine law can be used when the values of 2 sides on a triangle are given, and the angle in between them. It can also be used to find the angles in a triangle when the values of all three sides are given. Here’s the formula:
c2= a2+ b2 − 2ab cos(C)
Cosine Law
The above given formula can be used as a reference point to find different sides and different angles on the given triangle.
Side formulas:
a2= b2+ c2 − 2bc cos(A)
b2 = a2+ c2 − 2ac cos(B)
c2 = a2+ b2 − 2ab cos(C)
Angle formulas:
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
cos B =\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
cos C = \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
The cosine law is another form of the Pythagorean theorem.
Pythagorean theorem
Pythagorean theorem states that
c2 = a2+ b2
In the cosine law, give the angle a value of 90 degrees:
c2 = a2+ b2 − 2ab cos(90)
c2 = a2+ b2 − 0
c2 = a2+ b2
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Proof of Cosine Law
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Theorem: For any triangle with sides ABC and angles ABC,
c2 = a2+ b2 − 2ab cos(C)
To prove this, we’ll have to use the pythagorean theorem. This theorem states that, for any right angled triangle ABC,
c2 = a2+ b2
Cosine law can be used on all triangles. To use the Pythagorean theorem to prove the cosine law, we’ll have to divide a triangle with no right angles to make two right-angled triangles:
fig: Proof of Cosine Law
For triangle BCD, we can use the cosine function to find an expression for the CD. So,
cosC = \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
CD = acosC
By the Pythagorean theorem,
a2 = h2+ a2cos2c…….. (1)
Moving on to triangle BAD, we can find the value of DA with the value of CD;
DA = b - acosC
By the Pythagorean theorem,
c2 = h2+ (b - acosC)2
c2 = h2+b2 - 2(b)(a)cosC + a2cos2C………(2)
Subtract (1) from (2) or (2) - (1)
c2 - a2 = b2 - 2(b)(a)cosC
c2 = a2 + b2 - 2(b)(a)cosC
Hence Proved.
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When to Use Cosine Law
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The cosine law can be used when you are given:
- 3 sides or
- 2 sides and the angle between them
Cosine Law usage
For example, referring to the above triangle, you can use it when:
- all a, b, and c are known.
- a, c and angle B are known.
- b, c and angle A are known.
- b, a and angle C are known.
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Things to Remember
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- The law of cosines is a general form of the Pythagorean theorem and applies to all triangles.
- A specific set of details need to be known to be able to use the cosine law.
- The cosine law can be used to find both the sides of triangles and their internal angles.
- It can be proven by using the Pythagorean theorem.
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Sample Questions
Ques. Find the length of side a. (2 marks)
Ans. Using the cosine law,
a2 = (32)2 + (21)2 - 2(32)(21)cos40
a2 = 1024 + 441 - 1344(cos40)
a = \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
a = 20.87
Ques. Find angle B. (3 marks)
Ans. According to the cosine law,
192 = 232 + 272 - (23)(27)(cosB)
Rearrange that to make angle B the subject,
(23)(27)(cosB) = (232 + 272)/ 92
621(cosB) = 15.53
cosB = 15.53/621
cosB = 0.0250
cos-1B = 88.57
So, angle B = 88.57°
Ques. For triangle DEF, find the length of f given that d = 17 cm, e = 26 cm, and ∠F = 124°. (2 marks)
Ans. ∠F is in between sides d and e, and all their values are given. So the cosine law can be used here;
f2 = 172 + 262 - 2(17)(26)cos124
f2 = 1459.32
f = \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
f = 38.20 cm
Ques. Find the length of diagonal d in the parallelogram below. ∠ABC = 84° (3 marks)
Ans. Based on the rules of a parallelogram, we can first find ∠DAB.
∠DAB = 180 - ∠ABC = 180 -84 = 96°
∠DAB is the angle between AD, and AB, which along with d, form a triangle. Therefore, we can use the cosine law on triangle DAB to find the length of d.
d2 = 62 +122 - 2(6)(12)cos96
d2 = 195.05
d = 13.97
Ques. Find the missing values to their nearest whole number. (3 marks)
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
Ans. Internal angles in a triangle add up to 180 degrees, so we can find x
x = 180 - 34 - 115 = 31°
Use the sine law to find z
z/sin 34 = 13/sin115
z = (13/sin115) x sin34
z = 8.02 rounded off to 8
Use the cosine law to find y;
y2 = 132+ 82 - 2(13)(8)cos31
y2 = 54.71
y = 7.3 rounded off to 7
Ques. Find the missing angle below. (3 marks)
Ans. Using the cosine law,
5.92 = 3.12 + 4.32 - 2(3.1)(4.3)cosa
34.81 = 9.61 + 18.49 - 26.66cosa
26.66cosa = 9.61 + 18.49 - 34.81
cosa = (9.61 + 18.49 - 34.81)/ 26.66
cosa = -0.25
a = cos-1(-0.25)
a = 104.61°
Ques. Find the missing side below. (2 marks)
Ans: Using the cosine rule,
h2 = 882 + 1462 - 2(88)(146)cos53
h =\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
h = 116.6 rounded off to 116
Ques. A triangle has sides of length 22 m, 17 m, and 15 m. Find all the angles. (5 marks)
Ans. As all three sides are given, the cosine law can be used to find the angles.
Let the angle opposite side of length 22 m be x
222 = 172+152 - 2(17)(15)cosx
2(17)(15)cosx = ((172+152) - 222)
cosx = ((172+152) - 222)/ 2(17)(15)
x = cos-1((172+152) - 222)/ 2(17)(15)
x = 86.6°
Now that we know one set of sides and angle, we can use sine law to find another angle.
22/sin(86.6) = 17/sin(y)
22 sin(y) = 17 sin(86.6)
sin(y) = 17 sin(86.6)/ 22
y = sin-1(17 sin(86.6)/ 22)
y = 50.5°
We know two angles now, so we can find the remaining angle in the triangle by subtracting them from 180°.
z = 180 - 86.6 - 50.5 = 42.9°
Ques. In triangle ABC below, a = 15 cm, b = 14 cm, and ∠A = 44°. Find the value of c. (1 mark)
Ans: It can’t be found because we don’t have sufficient information. To use cosine law, we need the values of 2 sides and the angle between them. In this case, the value of the angle between the sides is not given and there is no means to find it.
Ques. Find a. (2 marks)
Ans. Using the cosine law,
a2 = 92 + 122 - 2(9)(12)cos25
a = \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
a = 5.4 ft
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