Trigonometry

Lesson

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 $BC=a$`B``C`=`a` , $CA=b$`C``A`=`b` and $AB=c$`A``B`=`c` and the opposite angles of the sides are respectively angle $A$`A`, angle $B$`B` and angle $C$`C`.

Law of cosines

$a^2=b^2+c^2-2bc\cos A$`a`2=`b`2+`c`2−2`b``c``c``o``s``A`

$b^2=a^2+c^2-2ac\cos B$`b`2=`a`2+`c`2−2`a``c``c``o``s``B`

$c^2=a^2+b^2-2ab\cos C$`c`2=`a`2+`b`2−2`a``b``c``o``s``C`

Notice that Pythagoras' Theorem $a^2=b^2+c^2$`a`2=`b`2+`c`2 makes an appearance in the Cosine Rule: $a^2=b^2+c^2-2bc\cos A$`a`2=`b`2+`c`2−2`b``c``c``o``s``A`

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 |

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.

Find the length of $a$`a` using the cosine rule.

Round your answer to two decimal places.

Find the value of angle $B$`B` using the cosine rule.

Round your answer to two decimal places.

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

Round your answer to two decimal places.