Introduction to Trigonometric Ratios

Trigonometry (or just 'trig') is everywhere around us. Engineers use it to design bridges, architects calculate roof slopes, and game developers work out character movements. The brilliant thing is, it all starts with simple right-angled triangles.

Before jumping into calculations, you've got to nail the labelling. Everything depends on which angle you're focusing on - we call this angle theta (written as θ). Get this wrong and everything else falls apart!

Quick Tip: Always identify your angle first, then label everything else relative to that angle.

The key is understanding that trigonometry only works with right-angled triangles - those with a perfect 90° corner.

Labelling Triangle Sides

Here's where students often trip up, but it's actually dead simple once you get it. You need to identify three sides relative to your chosen angle θ.

The Hypotenuse (H) is always the longest side - it's opposite the right angle and never changes. Easy to spot because it's the diagonal one.

The Opposite (O) side sits directly across from your angle θ. This one changes if you switch to looking at a different angle in the triangle.

The Adjacent (A) side is next to your angle θ (but it's not the hypotenuse). Like the opposite, this changes depending on which angle you're examining.

Remember: Opposite and Adjacent sides are always relative to your chosen angle. Switch angles, and they swap places!

The Three Main Trig Ratios

This is the heart of trigonometry - three simple ratios that connect angles to side lengths. The magic is that for any given angle, these ratios stay constant no matter how big or small your triangle is.

SOH CAH TOA is your best mate here - memorise it! It stands for:

• SOH: Sine = Opposite ÷ Hypotenuse
• CAH: Cosine = Adjacent ÷ Hypotenuse
• TOA: Tangent = Opposite ÷ Adjacent

These trigonometric ratios are the foundation of everything. Sine connects opposite and hypotenuse, cosine links adjacent and hypotenuse, whilst tangent relates opposite and adjacent.

Exam Tip: Write "SOH CAH TOA" at the top of your exam paper - it'll save you time and stress during questions!

Working with Given Triangles

Let's see SOH CAH TOA in action with a triangle that has sides of 5, 12, and 13, focusing on angle A.

First, identify your angle - we want angle A, so θ = A. Then label the sides: hypotenuse is 13 (longest side), opposite to A is 5, and adjacent to A is 12.

Now apply the ratios:

• sin(A) = 5/13 (opposite over hypotenuse)
• cos(A) = 12/13 (adjacent over hypotenuse)
• tan(A) = 5/12 (opposite over adjacent)

The brilliant thing is that these ratios would be exactly the same for any right-angled triangle with a matching angle, regardless of size.

Watch Out: If the question asked for angle B instead, your opposite and adjacent would swap, but the hypotenuse stays the same!

Finding Missing Side Lengths

Now for the really useful stuff - finding unknown sides using trigonometry. Say you've got a triangle with a 35° angle, hypotenuse of 15 cm, and you need to find the opposite side.

Start by identifying what you know: angle = 35°, hypotenuse = 15 cm, opposite = x (unknown). You don't need the adjacent for this problem.

Choose your ratio from SOH CAH TOA. You've got opposite and hypotenuse, so that's SOH - you need sine.

Set up your equation: sin(35°) = x/15. To find x, multiply both sides by 15: x = 15 × sin(35°).

Calculator Alert: Make sure your calculator is in DEG (degrees) mode, not RAD or GRAD - this catches loads of students out!

Solving and Key Points

Finishing the calculation: sin(35°) ≈ 0.57357, so x = 15 × 0.57357 ≈ 8.6 cm (to one decimal place).

Critical reminders that'll save your grades: SOH CAH TOA only works for right-angled triangles - no exceptions! Always check your calculator is in degrees mode before starting.

Labelling is everything - get your H, O, and A wrong and your whole answer goes wrong. The hypotenuse is always the longest side, which means sin and cos values are always less than 1.

Your problem-solving steps: label sides based on your angle, choose the right ratio, substitute values, solve for the unknown, and double-check that calculator mode!

Quick Check: If your sin or cos answer is greater than 1, something's gone wrong - probably your calculator mode or labelling!