Trigonometry Basics

Understanding trigonometry starts with remembering SOHCAHTOA - it's your best mate for solving triangle problems! This handy acronym tells you that sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, and tangent = opposite/adjacent.

When you're calculating angles, you'll use the inverse functions on your calculator. For example, if cos θ = 14/25, then θ = cos⁻¹(14/25) = 55.94°.

Quick Tip: Always check your calculator is in degree mode, not radians, unless the question specifically asks for radians!

The same method works for sin⁻¹ and tan⁻¹ - just make sure you're using the right ratio for the sides you know.

Calculating Sides and Pythagoras

Once you know an angle and one side, finding other sides becomes straightforward using CAH. If you need the adjacent side and know the hypotenuse, rearrange cos θ = A/H to get A = H × cos θ.

Pythagoras' theorem $a² + b² = c²$ is brilliant when you need to find the hypotenuse or when you don't know any angles. Remember that c is always the longest side - the hypotenuse.

Pro Tip: Sometimes it's better to leave answers as surds (like 2√41) rather than decimals, especially when the question asks for an exact answer.

Keep practicing both methods - trigonometry when you have angles, Pythagoras when you don't!

Understanding Surds

Surds are irrational numbers that can't be written as neat fractions - think √2 or √5. They're actually quite useful because they give you exact answers rather than messy decimals.

The key rule is √(a×b) = √a × √b, which lets you simplify surds by finding perfect square factors. For example, √50 = √(25×2) = 5√2.

When adding or subtracting surds, you can only combine like terms. So 2√125 - 3√80 becomes 10√5 - 12√5 = -2√5 after you've simplified each surd separately.

Remember: Always look for perfect square factors to simplify - it makes your final answer much neater and easier to work with.

Expanding Brackets with Surds

Expanding brackets with surds follows the same FOIL method you use with regular algebra, but you need to be extra careful with the surd multiplication. When you multiply surds together, √3 × √8 becomes √24, which you can then simplify further.

The trickiest bit is when surds multiply to give whole numbers. For instance, (5 + √3)(2 - √3) gives you √3 × √3 = 3, which simplifies your final answer significantly.

Watch Out: Always check if your surds can be simplified - √8 = 2√2, which makes your calculations much easier.

Take your time with these - one small mistake early on will mess up your entire answer!

Quadratics and Problem Solving

Quadratic equations pop up everywhere, so getting comfortable with expanding and factorising is crucial. When expanding triple brackets like $x+3$$x-2$$x-4$, do it in stages - expand two brackets first, then multiply by the third.

Problem solving with quadratics often involves setting up equations from real-world contexts. The key is translating the word problem into mathematical expressions, then solving as usual.

Strategy: Always check your answers make sense in the original context - negative lengths or impossible measurements are red flags!

Remember that quadratic equations usually have two solutions, but sometimes only one makes sense in the problem's context.

Indices and Powers

Memorising square numbers, cube numbers, and powers of small integers will save you tons of time in exams. Knowing that 144 = 12² or 243 = 3⁵ instantly makes calculations much faster.

The index laws are your toolkit for simplifying expressions: x^a × x^b = x^$a+b$, x^a ÷ x^b = x^$a-b$, and $x^a$^b = x^(ab). These rules work with any base, whether it's numbers or algebra.

Memory Trick: Practice these power sequences regularly - they'll become automatic and help you spot patterns in more complex problems.

Once you've got these basics down, you'll find that more advanced work with indices becomes much more manageable.