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Sine and cosine rules - Higher

Sine and cosine rules - Higher

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Explainer Video

Tutor: Bilal
Tutor: Bilal

Explainer Video 1

Explainer Video 2

Summary

Sine and cosine rules

In a nutshell

The sine and cosine rules can be used to work out a missing side or angle in any triangle. Similarly, the trigonometric area of a triangle can be used to find the area of a triangle without being given the perpendicular height.


Labelling a triangle

It is common convention to label the vertices (and corresponding angle) of a triangle with capital letters.  The side opposite the angle is labelled with its lowercase counterpart. Here is a diagram to illustrate:


Maths; Trigonometry; KS4 Year 10; Sine and cosine rules - Higher



The sine rule

When is it used?

The sine rule is used when you have an angle-side pair (e.g. side BBB​ and angle bbb​), and either another side or another angle.


The sine rule formula

There are two forms of the sine rule: the side form and the angle form.


Side form

Angle form

​asin⁡(A)=bsin⁡(B)=csin⁡(C)\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}sin(A)a​=sin(B)b​=sin(C)c​​​
​sin⁡(A)A=sin⁡(B)b=sin⁡(C)c\frac{\sin(A)}{A}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}Asin(A)​=bsin(B)​=csin(C)​​​

​

Example 1

In the diagram above, the angles AAA​ and BBB​ are 40∘40^\circ40∘ and 70∘70^\circ70∘ respectively. If the side aaa​ is 5cm5cm5cm​ long, what is the length of side bbb​ to the nearest integer?


You are given AAA​, BBB​, aaa and you are told to find the value of bbb. So, use the first 2 fractions of the sine rule. Use the side form as you need to find a length:

​asin⁡(A)=bsin⁡(B)\frac{a}{\sin(A)}=\frac{b}{\sin(B)}sin(A)a​=sin(B)b​​​


Substitute the known values:

​5sin⁡(40)=bsin⁡(70)\frac{5}{\sin(40)}=\frac{b}{\sin(70)}sin(40)5​=sin(70)b​


Solve for bbb​:​​

​b=sin⁡(70)×5sin⁡(40)=7.309511...b = \sin(70)\times \frac{5}{\sin(40)} = 7.309511...b=sin(70)×sin(40)5​=7.309511...​​


​b=7cm‾\underline{b =7cm}b=7cm​ to the nearest integer.

​

​

The cosine rule

When is it used?

The cosine rule is used when you have three sides or two sides and the angle between them.

​

The cosine rule formula

Again, there are two forms of the cosine rule: the side form and the angle form.


​​Side form

Angle form

​a2=b2+c2−2bccos⁡(A)a^2=b^2+c^2-2bc\cos(A)a2=b2+c2−2bccos(A)​​
​cos⁡(A)=b2+c2−a22bc\cos(A)=\frac{b^2+c^2-a^2}{2bc}cos(A)=2bcb2+c2−a2​​​


Tip: If you're confident enough in your algebra and rearranging, you can just memorise the side form.


Example 2

The lengths of the sides of a triangle are 5cm5cm5cm​, 7cm7cm7cm​, and 10cm10cm10cm​. What is the size of the angle opposite the side with length 7cm7cm7cm​ to 1 decimal place?


Always start off by drawing and labelling a sketch if you're not given one:


Maths; Trigonometry; KS4 Year 10; Sine and cosine rules - Higher


Note: Labelling correctly is very important when using the cosine rule. The angle form of the cosine rule has the angle AAA​, so make sure the side with length 7cm7cm7cm​ is labelled as aaa​. It doesn't matter which of the remaining sides you label bbb​ and ccc​.


Now, use the angle form of the cosine rule:

​cos⁡(A)=b2+c2−a22bc\cos(A)=\frac{b^2+c^2-a^2}{2bc}cos(A)=2bcb2+c2−a2​​​


Substitute in known values:

​cos⁡(A)=102+52−722(5)(10)\cos(A)=\frac{10^2+5^2-7^2}{2(5)(10)}cos(A)=2(5)(10)102+52−72​​​


​cos⁡(A)=0.76\cos(A)=0.76cos(A)=0.76​​


Use the inverse cosine function to get the angle by itself:

​A=cos⁡−1(0.76)=40.535802...A=\cos^{-1}(0.76)=40.535802...A=cos−1(0.76)=40.535802...​​


​A=40.5° (1 d.p.)‾\underline{A=40.5\degree \ (1 \ d.p.)}A=40.5° (1 d.p.)​



Area of a triangle

When is it used?

This is used when you have two sides and the angle between them, and want to find the area of the triangle.


The formula for the area of a triangle

​Area=12absin⁡(C)Area=\frac{1}{2}ab\sin(C)Area=21​absin(C)​​


Example 3

A triangle has sides 2cm2cm2cm​, 4cm4cm4cm​ and the angle between them is 24∘24^\circ24∘. Find the area of the triangle to 2 decimal places.


Sketch the triangle first and label the sides and angles.


Maths; Trigonometry; KS4 Year 10; Sine and cosine rules - Higher


Note: Again, it's important to label your triangle properly. The formula has sin⁡(C)\sin(C)sin(C), so you know that your given angle needs to be CCC​, which means the side opposite should be labelled as ccc​.


Now, use the formula:

​Area=12absin⁡(C)Area=\frac{1}{2}ab\sin(C)Area=21​absin(C)​​


Substitute in known values:

​Area=12(2)(4)sin⁡(24)Area = \frac{1}{2}(2)(4)\sin(24)Area=21​(2)(4)sin(24)​​


​Area=4×sin⁡(24)=1.62694....Area =4\times \sin(24)=1.62694....Area=4×sin(24)=1.62694....​​


​Area =1.63cm2 (2 d.p.)‾\underline{Area\,=1.63cm^2 \ (2 \ d.p.)}Area=1.63cm2 (2 d.p.)​


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Sine and cosine rules - Higher

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FAQs - Frequently Asked Questions

When can you use the trigonometric area of a triangle?

The trigonometric formula only works when you have two sides and the angle between them.

When can you use the sine rule and cosine rule?

The sine rule is used when you have an angle-side pair (e.g. side B and angle b), and another side or another angle. The cosine rule is used when you have 3 sides or two sides and the angle between them.

What is the main rule of labelling a triangle?

The angles must be capital letters and the opposite side must be its lowercase counterpart.

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