Understanding the Sine and Cosine Rules in Maths

Cosine Rule

When you're dealing with a triangle that's not right-angled, the cosine rule becomes your best mate. You'll use it when you know two sides and the angle between them, or when you know all three sides.

The formula is a² = b² + c² - 2bc cos A. If you need to find an angle instead, rearrange it to cos A = $b² + c² - a²$ / 2bc. Notice how it's like Pythagoras' theorem but with that extra bit tagged on the end.

Let's say you've got a triangle with sides 5cm and 10cm, and the angle between them is what you're after. If the opposite side is 9cm, you'd substitute into the rearranged formula. Work through: cos A = $5² + 10² - 9²$ / (2×5×10) = 44/100 = 0.44, then use cos⁻¹(0.44) to get 63.9°.

Quick tip: Remember to use the cos⁻¹ function on your calculator to convert back from the cosine value to the actual angle!

Sine Rule

The sine rule is brilliant for finding unknown angles or side lengths in non-right-angled triangles. It's especially handy when you know two angles and one side, or two sides and an angle opposite one of them.

The formula is a/sin A = b/sin B = c/sin C. Each side divided by the sine of its opposite angle gives the same value - pretty neat! You can rearrange this to find whatever you're looking for.

Here's how it works: if you've got a 6cm side opposite a 40° angle, and a 7cm side opposite angle C, set up 6/sin 40° = 7/sin C. Cross multiply to get sin C = (7 × sin 40°)/6 = 0.750. Then use sin⁻¹ to find C = 48.6°.

Watch out: Always remember to use the inverse sine function (sin⁻¹) to convert your decimal answer back to an angle - don't forget this crucial final step!

Practice Problem

Time to put your skills to the test! In triangle ABC, you've got angle A = 44°, angle C = 64°, and side BC = 4cm. Your mission is to find the length of AB and calculate the triangle's area.

Start by working out the missing angle B - remember all angles in a triangle add up to 180°. So angle B = 180° - 44° - 64° = 72°.

Now you can use the sine rule to find AB. Set up your equation: AB/sin C = BC/sin A, which gives you AB/sin 64° = 4/sin 44°. Cross multiply and solve to get AB to 3 decimal places.

For the area, use the formula Area = ½ab sin C, where a and b are two sides and C is the angle between them. You'll have all the values you need once you've found AB!

Pro tip: Always start by finding any missing angles first - it often makes the rest of the problem much easier to tackle!