Sine Rule (Leaving Cert Mathematics): Revision Notes

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Sine Rule

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 (AASAAS or ASAASA).
  • Two sides and a non-included angle (SSASSA).

The formula for the Sine Rule is:

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

Where:

  • aa, bb, and cc are the lengths of the sides of the triangle.
  • AA, BB, and CC are the angles opposite those sides.
infoNote

Important: When using the Sine Rule with SSA, there may be:

  • Two possible solutions (ambiguous case)
  • One solution
  • No solution

Always check for a second possible angle using: θ2=180∘−θ1\theta_2 = 180^\circ - \theta_1

The Sine Rule: Finding an Unknown Side

What Information do you need to be given?

  • Two angles and the length of a side.
image

What is the Formula?

The Sine Rule is given by:

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

Where a,ba, b, and cc are the sides of the triangle, and A,BA, B, and CC are the angles opposite these sides.

infoNote

Example:
Find the length of side xx in the triangle below where ∠P=37∘∠P=37^\circ, ∠Q=42∘∠Q=42^\circ, and side PQ=7.0 cmPQ=7.0 \text{ cm}.

imagexsin⁡42∘=7.0sin⁡37∘\frac{x}{\sin 42^\circ}=\frac{7.0}{\sin 37^\circ}

To find xx:

  1. Multiply both sides by sin⁡42∘\sin 42^\circ:
x=7.0×sin⁡42∘sin⁡37∘x=\frac{7.0 \times \sin 42^\circ}{\sin 37^\circ}
  1. Substitute the values:
x=7.0×0.66910.6018≈6.3 cm  (1 dp)x=\frac{7.0 \times 0.6691}{0.6018}\approx 6.3 \text{ cm} \; (1\text{ dp})

So, the length of side xx is approximately 6.3 cm.

The Sine Rule: Finding an Unknown Angle

What Information do you need to be given?

  • Two sides and an angle not between them (SSA).
image

What is the Formula?

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

sin⁡Aa=sin⁡Bb=sin⁡Cc\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}
infoNote

Example:
Find the size of angle xx in the triangle below where a=11 cma=11 \text{ cm}, b=16 cmb=16 \text{ cm}, and ∠B=37∘∠B=37^\circ.

imagesin⁡x16=sin⁡37∘11\frac{\sin x}{16}=\frac{\sin 37^\circ}{11}

To find sin⁡x\sin x:

  1. Multiply both sides by 1616:
sin⁡x=16×sin⁡37∘11\sin x=\frac{16 \times \sin 37^\circ}{11}
  1. Substitute the values:
sin⁡x=16×0.601811≈0.8753\sin x=\frac{16 \times 0.6018}{11}\approx 0.8753
  1. To find xx, take the inverse sine:
x=sin⁡−1(0.8753)≈61.1∘  (1 dp)x=\sin^{-1}(0.8753)\approx 61.1^\circ \; (1\text{ dp})

So, the angle xx is approximately 61.1°.

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