We have already seen the Sine Rule, and Cosine Rule in action.
The sine rule (or the law of sines) states that the ratio of the sine of any angle to the length of the side opposite that angle is the same for all three angles of a triangle
$\frac{\sin A}{a}$sinAa= $\frac{\sin B}{b}=\frac{\sin C}{c}$sinBb=sinCc
It can be used, for example, when two angles and a side are known, such as in triangulation - a process for finding the location of a point.
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:
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.
$a^2=b^2+c^2-2bc\cos A$a2=b2+c2−2bccosA
$b^2=a^2+c^2-2ac\cos B$b2=a2+c2−2accosB
$c^2=a^2+b^2-2ab\cos C$c2=a2+b2−2abcosC
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+c2−2bccosA
The questions in this set will require you to choose whether you are needing to use the Sine Rule or the Cosine Rule, (or indeed other aspects of trigonometry).
Use the Sine Rule
Use the Cosine Rule
Calculate the length of $y$y in metres.
Round your answer to one decimal place.
Find the value of angle $w$w in degrees.
Round your answer to two decimal places.
$\triangle ABC$△ABC consists of angles $A$A, $B$B and $C$C which appear opposite sides $a$a, $b$b and $c$c respectively. Consider the case where the measures of $a$a, $c$c and $A$A are given.
Which of the following is given?
$SSA$SSA: Two sides and an angle
$SAS$SAS: Two sides and the included angle
$SAA$SAA: two angles and a side
$ASA$ASA: two angles and the side between them
$SSS$SSS: Three sides
Which law should be used to start solving the triangle?
the law of sines
the law of cosines