The Law of Cosines relates the lengths of the sides of a triangle with the cosine of one of its angles.
The Law of Cosines, for any triangle ABC is
a 2 = b 2 + c 2 – 2bc cos A
b 2 = a 2 + c 2 – 2ac cos B
c 2 = a 2 + b 2 – 2ab cos C
The Law of Cosines is also sometimes called the Cosine Rule or Cosine Formula.
If we are given two sides and an included angle (SAS) or three sides (SSS) then we can use the Law of Cosines to solve the triangle ie. to find all the unknown sides and angles.
We can use the Law of Sines to solve triangles when we are given two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA).
Solve triangle PQR in which p = 6.5 cm, q = 7.4 cm and ∠R = 58°.
Using the Cosine rule,
r2 = p2 + q2– 2pq cos R
r2 = (6.5)2 + (7.4)2– 2(6.5)(7.4) cos58°
r = 6.78 cm
Using the Sine rule,
∠Q = 180° – 58° – 54.39°
∠P = 54.39°, ∠Q = 67.61° and r = 6.78 cm
In triangle ABC, a = 9 cm, b = 10 cm and c = 13 cm. Find the size of the largest angle.
The largest angle is the one facing the longest side, i.e. C.
c2 = a2 + b2– 2ab cos C
∠C = 86.2°
The video shows how to solve two word problems using the Law of Cosines
Example 1: An engineering firm decides to bid on a proposed tunnel through a mountain. Find how long is the tunnel and the bid amount.Example 2: Find the range of service of a transmission tower.
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