Sine and Cosine rules Revision Notes for OCR GCSE Maths

Sine and Cosine Rules

Introduction to Sine and Cosine Rules

In trigonometry, the Sine and Cosine Rules are incredibly powerful tools that extend the basic trigonometric ratios (Sin, Cos, Tan) to any triangle—not just right-angled triangles. This makes them extremely versatile and useful for solving a wide variety of problems.

The Sine Rule

The Sine Rule is used when you know:

Two angles and one side ( $AAS$ or $ASA$ ).
Two sides and a non-included angle ( $SSA$ ). The formula for the Sine Rule is:

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

Where:

$a$ , $b$ , and $c$ are the lengths of the sides of the triangle.
$A$ , $B$ , and $C$ are the angles opposite those sides.

1. The Sine Rule: Finding an Unknown Side

What Information do you need to be given?

Two angles and the length of a side.

What is the Formula?

The Sine Rule is given by:

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

Where $a, b$ , and $c$ are the sides of the triangle, and $A, B$ , and $C$ are the angles opposite these sides.

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Example: Find the length of side x_x_ in the triangle below where $∠P=37\degree$ , $∠Q=42\degree$ , and side $PQ=7.0 cm$ .

\frac{x}{sin⁡\ 42\degree}=\frac{7.0}{sin\ ⁡37\degree}

To find $x$ :

Multiply both sides by $sin42\degree$ :

x=\frac{7.0×sin\ 42\degree}{sin \ 37\degree}

Substitute the values:

x=\frac{7.0×0.6691}{0.6018}≈6.3\ cm(1dp)

So, the length of side $x$ is approximately 6.3 cm.

2. The Sine Rule: Finding an Unknown Angle

What Information do you need to be given?

Two lengths of sides and the angle not included (i.e., not between those two sides).

What is the Formula?

The Sine Rule can also be rearranged to find an unknown angle:

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

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Example: Find the size of angle x_x_ in the triangle below where $a=11 \ cm$ , $b=16 \ cm$ , and $∠B=37\degree$ .

\frac{sin⁡\ x}{16}=\frac{sin⁡37\degree}{11}

To find $sin\ ⁡x$ :

Multiply both sides by $16$ :

sin\ ⁡x=\frac{16×sin⁡\ 37\degree}{11}

Substitute the values:

Sin⁡\ x=\frac{16×0.6018}{11}≈0.8753

To find $x$ , take the inverse sine:

x=sin⁡^{−1}(0.8753)≈61.1\degree (1dp)

So, the angle $x$ is approximately 61.1°.

3. The Cosine Rule: Finding an Unknown Side

What Information do you need to be given?

Two sides of the triangle and the included angle (i.e., the angle between the two sides).

What is the Formula?

The Cosine Rule for finding an unknown side is:

a^2=b^2+c^2−2bc⋅cos⁡A

Where:

$a$ , $b$ , and $c$ are the sides of the triangle.
$A$ is the angle opposite side $a$ .

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Example: Find the length of side $x$ in the triangle where $b=5.2 m$ , $c=4.5m$ , and $∠A=58\degree$ .

Substitute the values into the formula:

x^2=5.2^2+4.52−2×5.2×4.5×cos⁡58\degree

Calculate each term:

x^2=27.04+20.25−23.4×0.5299

x^2=22.48977 m^2

Square root both sides:

x=\sqrt{22.48977}≈4.74 \ m (2dp)

So, the length of side $x$ is approximately 4.74 m.

4. The Cosine Rule: Finding an Unknown Angle

What Information do you need to be given?

All three lengths of the triangle.

What is the Formula?

The Cosine Rule for finding an unknown angle is:

cos\ ⁡A=\frac{b^2+c^2−a^2}{2bc}

Where:

$a, b$ , and $c$ are the sides of the triangle.
$A$ is the angle opposite side $a$ .

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Example: Find the size of angle $x$ in the triangle where $a=9 m$ , $b=11 m$ , and $c=12 m$ .

Substitute the values into the formula:

cos⁡\ x=\frac{11^2+12^2−9^2}{2×11×12}

Calculate each term:

cos\ ⁡x=\frac{121+144−81}{264}=\frac{184}{264}≈0.69697

Use the inverse cosine to find $x$ :

x=cos^{⁡−1}(0.69697)≈72.97\degree (2dp)

So, the angle is approximately 72.97°

Summary

The Sine Rule

The Sine Rule is used when we are given either:

Two angles and one side ( $AAS$ or $ASA$ )
Two sides and a non-included angle ( $SSA$ ) Formula:

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

Where:

$a,b,c$ are the sides of the triangle
$A,B,C$ are the angles opposite those sides Finding an Unknown Side:

When we have two angles and one side, we can find the unknown side using the formula:

\frac{x}{sin⁡\ X}=\frac{7.0}{sin⁡\ 42\degree}

Worked Example:

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Example

Given:

Angle $X=42\degree$
Side opposite angle $X = 7.0 cm$
Another angle $= 37°$ To find the unknown side $x$ :

x=\frac{7.0×sin\ ⁡37\degree}{sin\ ⁡42\degree}≈6.3\ cm

Finding an Unknown Angle:

When given two sides and an angle not included between them, the Sine Rule can help find the unknown angle.

Given:

Sides $a=16 cm$ and $b=11 cm$
Angle $B=37\degree$ To find angle $A$ :

sin\ ⁡A=\frac{16×sin⁡\ 37\degree}{11}

Then use the inverse sine to find $A$ :

A=sin⁡^{−1}(0.8753)≈61.1\degree

The Cosine Rule

The Cosine Rule is used when we are given:

Three sides ( $SSS$ )
Two sides and the included angle (SAS) Formula for Finding a Side:

c^2=a^2+b^2−2\ abcos\ ⁡C

Where:

$a,b,c$ are the sides of the triangle
$C$ is the included angle between sides $a$ and $b$

Worked Example:

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Example

Given:

Side $a=5.2 cm$ , side $b=4.5 cm$
Included angle $C=58\degree$ To find the side $c$ :

c^2=5.2^2+4.5^2−2×5.2×4.5×cos\ ⁡58\degree

≈4.74 cm

Formula for Finding an Angle:

If all three sides of the triangle are known, the Cosine Rule helps find an unknown angle.

Example:

Given:

Sides $a=9 cm, b=11 cm$ , and $c=12 cm$

To find angle $A$ :

cos⁡\ A=\frac{b^2+c^2−a^2}{2bc}

A=cos^{⁡−1}(0.292929)≈72.97\degree

Sine and Cosine rules (OCR GCSE Maths): Revision Notes