 Cosine Law

Lesson

Cosine Rule

In trigonometry the cosine rule relates the lengths of the sides and the cosine of one of its angles.

The Law of Cosines is useful in finding:

• the third side of a triangle when you know two sides and the angle between them
• the angles of a triangle when you know all three sides ABC is a triangle with side lengths $BC=a$BC=a , $CA=b$CA=b and $AB=c$AB=c and the opposite angles of the sides are respectively angle $A$A, angle $B$B and angle $C$C.

Law of cosines

$a^2=b^2+c^2-2bc\cos A$a2=b2+c22bccosA

$b^2=a^2+c^2-2ac\cos B$b2=a2+c22accosB

$c^2=a^2+b^2-2ab\cos C$c2=a2+b22abcosC

Notice that Pythagoras' Theorem $a^2=b^2+c^2$a2=b2+c2 makes an appearance in the Cosine Rule: $a^2=b^2+c^2-2bc\cos A$a2=b2+c22bccosA

Finding an angle

Find angle $B$B in the triangle. Think: All three side lengths are known, so I can apply the cosine rule. The unknown angle B appears opposite side $b=3$b=3.

 $b^2$b2 $=$= $a^2+c^2-2ac\cos B$a2+c2−2accosB $3^2$32 $=$= $5^2+6^2-2\times5\times6\cos B$52+62−2×5×6cosB $9$9 $=$= $25+36-60\cos B$25+36−60cosB $9-61$9−61 $=$= $-60\cos B$−60cosB $\frac{-52}{-60}$−52−60​ $=$= $\cos B$cosB $\cos B$cosB $=$= $0.866667$0.866667 $B$B $=$= $29.9^\circ$29.9°  to $1$1 decimal place

Finding a side length Find the value of $x$x in the diagram.

Think: The first thing I always do is identify which side is opposite the given angle. This side is the subject of the formula. To find out which other values we are given I label the sides and angles using $a$a,$b$b and $c$c .

Do

I add the following labels to the triangle: So I want to find the value of $c$c.

 $c^2$c2 $=$= $a^2+b^2-2ab\cos C$a2+b2−2abcosC $c^2$c2 $=$= $8^2+11^2-2\times8\times11\cos39^\circ$82+112−2×8×11cos39° $c^2$c2 $=$= $64+121-176\cos39^\circ$64+121−176cos39° $c^2$c2 $=$= $48.22$48.22 $c$c $=$= $6.94$6.94

The following interactive demonstrates that the cosine rule holds regardless of the angles or size and shape of the triangle.

Examples

Question 1

Find the length of $a$a using the cosine rule. Question 2

Find the value of angle $B$B using the cosine rule. Question 3

Find the length of $c$c using the cosine rule. Outcomes

10D.T3.02

Explore the development of the cosine law within acute triangles

10D.T3.03

Determine the measures of sides and angles in acute triangles, using the sine law and the cosine law