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Maths349Updated 29 Sept 20264 pages

Trigonometry Sine and Cosine Rules for Non-Calculator Problems

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Emilia Wuss@xed7even
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The sine and cosine rules are powerful tools for solving non-right-angled triangles when you don't have a 90° angle to work with. These formulas help you find missing sides and angles in any triangle - skills you'll definitely need for your GCSE maths exam.
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The Sine Rule

Ever wondered how to solve triangles that aren't right-angled? The sine rule is your go-to formula when dealing with any triangle where you know some sides and angles.

The sine rule states: asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}. This can also be flipped to: sin⁡Aa=sin⁡Bb=sin⁡Cc\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}.

Looking at the example: to find side 'a' when you know angle A = 37°, side b = 7.9, and angle B = 83°, you substitute into the formula. You get a=7.9×sin⁡(37°)sin⁡(83°)a = \frac{7.9 \times \sin(37°)}{\sin(83°)}, which gives you the missing side length.

Remember: Use the sine rule when you have two angles and one side, or two sides and a non-included angle.

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When to Use the Sine Rule

The sine rule works brilliantly in specific situations, and knowing when to use it will save you loads of time in exams.

Use the sine rule when you have two angles and one side (AAS or ASA triangles), or when you have two sides and an angle opposite one of them (SSA). The example shows angle A = 29°, side a = 19.2cm, and side c = 17.1cm.

Set up the equation: sin⁡(29°)19.2=sin⁡(C)17.1\frac{\sin(29°)}{19.2} = \frac{\sin(C)}{17.1}. Cross-multiply to find the unknown angle C. This method is straightforward once you've identified which parts of the triangle you know.

Top tip: Always label your triangle clearly before starting - it prevents mix-ups with sides and angles!

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The Cosine Rule - When to Use It

Sometimes the sine rule just won't cut it, and that's where the cosine rule becomes your best mate in triangle problems.

Use the cosine rule when you have a non-right-angled triangle and need to find a length with two known lengths plus the angle between them. You'll also use it when you know all three sides and need to find an angle.

The cosine rule is perfect for those tricky triangles where the sine rule leaves you stuck. If you've got two sides and the included angle, or three sides with no angles, the cosine rule is your answer.

Quick check: If you can't use the sine rule, try the cosine rule - one of them will always work!

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The Cosine Rule Formula

The cosine rule formula is: a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A. Notice how it's similar to Pythagoras' theorem but with an extra bit tacked on.

This formula helps you find the length of side 'a' when you know the other two sides (b and c) and the angle opposite to side 'a' (angle A). You can rearrange this formula for different sides depending on what you're looking for.

The cosine rule is incredibly reliable and works for any triangle - acute, obtuse, or right-angled. When the angle is 90°, cos(90°) = 0, so the formula actually becomes Pythagoras' theorem!

Pro tip: The cosine rule is your safety net - it works when other methods fail, making it perfect for exam confidence.

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