Master SOH CAH TOA - the key to solving any right-angled triangle!
The Most Important Memory Aid in GCSE Maths!
SOH
Sin = Opposite / Hypotenuse
CAH
Cos = Adjacent / Hypotenuse
TOA
Tan = Opposite / Adjacent
Trigonometry is the study of relationships between sides and angles in RIGHT-ANGLED triangles. While Pythagoras tells us about sides, trigonometry connects sides with angles. This opens up a whole new world of problem-solving!
Labeling the Triangle (from angle θ):
🔴 SINE (sin θ)
Compares the OPPOSITE side to the HYPOTENUSE. Use when you have/need O and H.
🟢 COSINE (cos θ)
Compares the ADJACENT side to the HYPOTENUSE. Use when you have/need A and H.
🔵 TANGENT (tan θ)
Compares the OPPOSITE side to the ADJACENT. Use when you have/need O and A (no hypotenuse!).
Trigonometry questions appear in EVERY GCSE paper. Worth many marks - you MUST master SOH CAH TOA!
Used in navigation, engineering, architecture, astronomy, and physics. Essential for A-level sciences!
Allows you to find missing sides and angles when Pythagoras alone isn't enough. Game-changing skill!
Question: In a right-angled triangle, the hypotenuse is 10cm and angle θ = 30°. Find the opposite side.
Step 1: Identify: We have H=10, angle=30°, need O
Step 2: Choose: Sin (we have H and need O)
Step 3: Formula: sin(30°) = O/10
Step 4: Rearrange: O = 10 × sin(30°)
Step 5: Calculate: O = 10 × 0.5 = 5cm
Answer: Opposite = 5cm
Question: A right-angled triangle has hypotenuse 12cm and angle 60°. Find the adjacent side.
Step 1: Identify: H=12, angle=60°, need A
Step 2: Choose: Cos (we have H and need A)
Step 3: Formula: cos(60°) = A/12
Step 4: Rearrange: A = 12 × cos(60°)
Step 5: Calculate: A = 12 × 0.5 = 6cm
Answer: Adjacent = 6cm
Question: The adjacent side is 8cm and angle is 35°. Find the opposite side.
Step 1: Identify: A=8, angle=35°, need O
Step 2: Choose: Tan (we have A and need O)
Step 3: Formula: tan(35°) = O/8
Step 4: Rearrange: O = 8 × tan(35°)
Step 5: Calculate: O = 8 × 0.7002 = 5.6cm
Answer: Opposite ≈ 5.6cm
Question: You have angle=45°, opposite=7cm, hypotenuse=?. Which ratio?
Step 1: Identify what you have: O and angle
Step 2: Identify what you need: H
Step 3: O and H → Use SINE!
Step 4: sin(45°) = 7/H
Step 5: H = 7 ÷ sin(45°) = 9.9cm
Answer: Use SIN, H = 9.9cm
Question: A 15m ladder leans against a wall at 70° to the ground. How high up the wall does it reach?
Step 1: Draw right-angled triangle
Step 2: H=15m (ladder), angle=70°, need O (height)
Step 3: Use: sin(70°) = O/15
Step 4: O = 15 × sin(70°)
Step 5: O = 15 × 0.9397 = 14.1m
Answer: Height = 14.1m
Master SOH CAH TOA with these problems:
What students often do wrong:
Students confuse which side is opposite and which is adjacent, especially if the angle isn't at the bottom left!
Correct approach: Always label from YOUR angle: Opposite = ACROSS from angle, Adjacent = NEXT TO the angle (touching it). The hypotenuse never changes - always longest!
Memory tip: "Stand at your angle - opposite is across, adjacent is beside you"
What students often do wrong:
Calculator is in RADIANS mode instead of DEGREES mode, giving completely wrong answers!
Correct approach: ALWAYS check your calculator is in DEGREES (DEG) mode. Test: sin(30) should give 0.5. If not, change mode!
Memory tip: "DEG for degrees - check before every question!"
What students often do wrong:
Students randomly pick sin, cos, or tan without checking which sides they have/need.
Correct approach: Always follow these steps: 1) Label O, A, H. 2) Identify which TWO you have/need. 3) Choose the ratio with those two letters.
Memory tip: "Label first, choose second, calculate third!"
Show ALL your working! Write down which ratio you're using, show the substitution, and show each calculation step. This can earn you method marks even if your final answer is wrong!
You've unlocked Trigonometry - one of the most powerful tools in GCSE Maths!
This memory aid will serve you throughout GCSE and beyond!
Next: Finding sides using trigonometry (more practice and harder problems!)