Sine and Cosine RuleIntroductionThis section will cover how to: Use the Sine Rule to find unknown sides and angles Use the Cosine Rule to find unknown sides and angles Combine trigonometry skills to solve problemsEach topic is introduced with a theory section including examples and then some practice questions. At the end of the page there is an exercise where you can test your understanding of all the topics covered in this page.You are allowed to use calculators in this topic.All answers should be given to 3 significant figures unless otherwise stated.Formulae You Should KnowYou should already know each of the following formulae:formulae for right-angled triangles

formulae for all triangles

NOTE: The only formula above which is in the A Level Maths formula book is the one highlighted in yellow.You must learn these formulae, and then try to complete this page without referring to the table above.Sine RuleThe Sine Rule can be used in any triangle (not just right-angled triangles) where a side and its opposite angle are known.Finding SidesIf you need to find the length of a side, you need to use the version of the Sine Rule where the lengths are on the top:a=b

sin(A)sin(B)

You will only ever need two parts of the Sine Rule formula, not all three.You will need to know at least one pair of a side with its opposite angle to use the Sine Rule.

Finding Sides ExampleWork out the length ofxin the diagram below:

Step 1Start by writing out the Sine Rule formula for finding sides:a=b

sin(A)sin(B)

Step 2Fill in the values you know, and the unknown length:x=7

sin(80)sin(60)

Remember that each fraction in the Sine Rule formula should contain a side and its opposite angle.

Step 3Solve the resulting equation to find the unknown side, giving your answer to 3 significant figures:x=7(multiply by sin(80) on both sides)

sin(80)sin(60)

x=7 sin(80)

sin(60)

x=7.96 (accurate to 3 significant figures)

Note that you should try and keep full accuracy until the end of your calculation to avoid errors.

Finding AnglesIf you need to find the size of an angle, you need to use the version of the Sine Rule where the angles are on the top:sin(A)=sin(B)

ab

As before, you will only need two parts of the Sine Rule , and you still need at least a side and its opposite angle.

Finding Angles ExampleWork out anglem in the diagram below:

Step 1Start by writing out the Sine Rule formula for finding angles:sin(A)=sin(B)

ab

Step 2Fill in the values you know, and the unknown angle:sin(m)=sin(75)

810

Remember that each fraction in the Sine Rule formula should contain a side and its opposite angle.

Step 3Solve the resulting equation to find the sine of the unknown angle:sin(m)=sin(75)(multiply by 8 on both sides)

810

sin(m)=sin(75) 8

10

sin(m)=0.773 (3 significant figures)

Step4Use the inverse-sine function (sin1) to find the angle:m=sin1(0.773) = 50.6 (3sf)

Other NotesYou may be aware that sometimes Sine Rule questions can have two solutions (only when you are finding angles) you do not need to know about these additional solutions at this time but you will learn more about them next year.

Practice QuestionsWork out the answer to each question then click on the button markedto see if you are correct.

(a) Find the missing side in the diagram below:

(b) Find the missing angle in the diagram below:

Cosine RuleThe Cosine Rule can be used in any triangle where you are trying to relate all three sides to one angle.Finding SidesIf you need to find the length of a side, you need to know the other two sides and the opposite angle.You need to use the version of the Cosine Rule wherea2is the subject of the formula:a2=b2+c2 2bccos(A)

Sideais the one you are trying to find. Sidesbandcare the other two sides, and angleAis the angle opposite sidea.

Finding Sides ExampleWork out the length ofxin the diagram below:

Step 1Start by writing out the Cosine Rule formula for finding sides:a2=b2+c2 2bccos(A)

Step 2Fill in the values you know, and the unknown length:x2= 222+ 282 22228cos(97)

It doesn't matter which way around you put sidesbandc it will work both ways.

Step 3Evaluate the right-hand-side and then square-root to find the length:x2=222+ 282 22228cos(97)(evaluate the right hand side)

x2=1418.143.....(square-root both sides)

x=37.7 (accurate to 3 significant figures)

As with the Sine Rule you should try and keep full accuracy until the end of your calculation to avoid errors.

Finding AnglesIf you need to find the size of an angle, you need to use the version of the Cosine Rule where the cos(A) is on the left:cos(A)=b2+c2a2

2bc

It is very important to get the terms on the top in the correct order;bandcare either side of angleAwhich you are trying to find and these can be either way around, but sideamust be the side opposite angleA.

Finding Angles ExampleWork out angleP in the diagram below:

Step 1Start by writing out the Cosine Rule formula for finding angles:cos(A)=b2+c2a2

2bc

Step 2Fill in the values you know, and the unknown length:cos(P)=52+ 82 72

2 5 8

Remember to make sure that the terms on top of the fraction are in the correct order.

Step 3Evaluate the right-hand-side and then use inverse-cosine (cos1) to find the angle:cos(P)=52+ 82 72(evaluate the right-hand side)

2 5 8

cos(P)=0.5(do the inverse-cosine of both sides)

P=cos1(0.5) = 60 (3sf)

Other NotesIf you know two sides and an angle which is not inbetween them then you can use the Cosine Rule to find the other side, but it is easier to use the Sine Rule in this situation you should always use the Sine Rule if you have an angle and its opposite side