Beth Clarke

11/27/2025

Maths

sine and cosine rule

Subjects

Maths

Law of Sines & Cosines

Law of Sines

102

•

Nov 27, 2025

•

3 pages

Understanding Sine and Cosine Rules

Beth Clarke

@bethclarke_pbon

The sine rule and cosine rule are essential mathematical tools... Show more

1 / 3

$### Sine rule $\frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{sinc}$ $\frac{SinA}{a} = \frac{SinB}{b} = \frac{Sin C}{c}$$

Sine Rule

The sine rule establishes a relationship between the sides and angles of any triangle. It can be written in two equivalent ways:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Or alternatively:

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

This powerful rule is useful when you know either:

Two angles and one side (AAS or ASA)
Two sides and the angle opposite one of them (SSA)

💡 Think of the sine rule as matching pairs - each side with its opposite angle. The rule works for any triangle, not just right-angled ones!

$### Sine rule $\frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{sinc}$ $\frac{SinA}{a} = \frac{SinB}{b} = \frac{Sin C}{c}$$

Sine Rule Example

Let's see how to apply the sine rule to find an unknown side length:

Given a triangle with:

A side length of 15 units
The angle opposite this side is 65°
Another angle in the triangle is 31°

We can rearrange the sine rule to find the unknown side x:

$\frac{15}{\sin 65°} = \frac{x}{\sin 31°}$

Multiplying both sides by $\sin 31°$ :

$x = \frac{15 \times \sin 31°}{\sin 65°} = 26.40$

The unknown side length is 26.40 units.

$### Sine rule $\frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{sinc}$ $\frac{SinA}{a} = \frac{SinB}{b} = \frac{Sin C}{c}$$

Cosine Rule Example

The cosine rule is perfect when you have:

Three sides and need an angle
Two sides and the included angle (to find the third side)

In this example, we have a triangle with sides 15.5 cm, 21.8 cm, and 23.6 cm, and we need to find angle θ.

The cosine rule states: $\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}$

Substituting our values: $\cos(\theta) = \frac{15.5^2 + 23.6^2 - 21.8^2}{2 \times 15.5 \times 23.6} = \frac{321.97}{731.6} = 0.4400...$

Taking the inverse cosine: $\theta = \cos^{-1}(0.4400) = 63.89°$

🔍 Remember: The cosine rule is especially useful when you don't have enough information to use the sine rule directly!

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Anna

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Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Subjects

Maths

Law of Sines & Cosines

Law of Sines

Maths

•

102

•

Nov 27, 2025

•

3 pages

Understanding Sine and Cosine Rules

Beth Clarke

@bethclarke_pbon

The sine rule and cosine rule are essential mathematical tools for solving triangles when you don't have enough information to use basic trigonometry. These rules allow you to find missing sides or angles in any triangle, not just right-angled ones.

$### Sine rule $\frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{sinc}$ $\frac{SinA}{a} = \frac{SinB}{b} = \frac{Sin C}{c}$$

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Sine Rule

The sine rule establishes a relationship between the sides and angles of any triangle. It can be written in two equivalent ways:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Or alternatively:

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

This powerful rule is useful when you know either:

Two angles and one side (AAS or ASA)
Two sides and the angle opposite one of them (SSA)

💡 Think of the sine rule as matching pairs - each side with its opposite angle. The rule works for any triangle, not just right-angled ones!

$### Sine rule $\frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{sinc}$ $\frac{SinA}{a} = \frac{SinB}{b} = \frac{Sin C}{c}$$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Sine Rule Example

Let's see how to apply the sine rule to find an unknown side length:

Given a triangle with:

A side length of 15 units
The angle opposite this side is 65°
Another angle in the triangle is 31°

We can rearrange the sine rule to find the unknown side x:

$\frac{15}{\sin 65°} = \frac{x}{\sin 31°}$

Multiplying both sides by $\sin 31°$ :

$x = \frac{15 \times \sin 31°}{\sin 65°} = 26.40$

The unknown side length is 26.40 units.

$### Sine rule $\frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{sinc}$ $\frac{SinA}{a} = \frac{SinB}{b} = \frac{Sin C}{c}$$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Cosine Rule Example

The cosine rule is perfect when you have:

Three sides and need an angle
Two sides and the included angle (to find the third side)

In this example, we have a triangle with sides 15.5 cm, 21.8 cm, and 23.6 cm, and we need to find angle θ.

The cosine rule states: $\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}$

Substituting our values: $\cos(\theta) = \frac{15.5^2 + 23.6^2 - 21.8^2}{2 \times 15.5 \times 23.6} = \frac{321.97}{731.6} = 0.4400...$

Taking the inverse cosine: $\theta = \cos^{-1}(0.4400) = 63.89°$

🔍 Remember: The cosine rule is especially useful when you don't have enough information to use the sine rule directly!

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Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Understanding Sine and Cosine Rules

Sine Rule

Sine Rule Example

Cosine Rule Example

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