Basic Trigonometry

GCSE Higher trigonometry covers how the angles inside a right-angled triangle fix the exact ratio of its side lengths. Managing how to label the sides, select the correct ratio, and remember the non-calculator exact values is critical for core exam marks.

a) Labelling the triangle sides:

The three sides of a right-angled triangle are labelled dynamically based on the position of the target angle θ.
Hypotenuse (H)  →  The longest side, always directly opposite the 90° right angle.
Opposite (O)  →  The side directly across from the chosen angle θ.
Adjacent (A)  →  The side next to the angle θ that is not the hypotenuse.
At GCSE, mislabelling the adjacent and opposite sides is the most common reason for losing full marks on a multi-step problem.

Note!
Although θ is common in basic trig questions, for the more advanced situations requiring the sine, cosine, or area formulas, the sides are labelled differently - this time they are labelled the same was as in pythagoras, where lower-case 'c' is the hypotenuse and 'a' and 'b' can be either of opposite or adjacent.

In this system, the angles are labelled in *upper case* so A B or C, and the angle letter is *always* opposite the side with the same lower-case letter.

b) Selecting the formula (SOH CAH TOA):

The three core trigonometric ratios define the relationships between the sides and the angle θ.
sin θ = O ÷ H      cos θ = A ÷ H      tan θ = O ÷ A

Finding a missing side example: Angle θ = 30°, Hypotenuse = 12cm. Find the opposite side.
Identify sides  →  Given H, want O  →  Select SOH  →  sin θ = O ÷ H
Rearrange  →  O = H × sin θ  →  O = 12 × sin 30
Substitute value  →  O = 12 × 1/2 = 6cm

Finding a missing angle example: Opposite = 5cm, Adjacent = 5cm. Find the angle θ.
Identify sides  →  Given O and A  →  Select TOA  →  tan θ = O ÷ A
Substitute values  →  tan θ = 5 ÷ 5 = 1
Inverse function  →  θ = tan⁻¹(1) = 45°


c) Exact trig values (Non-Calculator):

You must memorize the exact outputs for the angles 0°, 30°, 45°, 60°, and 90°.

Sine values:
sin 0° = 0      sin 30° = 1/2      sin 45° = 1/√2      sin 60° = √3/2      sin 90° = 1

Cosine values:
cos 0° = 1      cos 30° = √3/2      cos 45° = 1/√2      cos 60° = 1/2      cos 90° = 0

Tangent values:
tan 0° = 0      tan 30° = 1/√3      tan 45° = 1      tan 60° = √3      tan 90° = Undefined


d) Key non-calculator surd triangles:

All non-calculator exact values are derived from two geometric base triangles scaled up or down.
The 45° Triangle (Isosceles):
Has sides where Opposite = 1, Adjacent = 1, and Hypotenuse = √2.
The 30° and 60° Triangle (Half an Equilateral):
Has sides where Hypotenuse = 2, Shorter side = 1, and Mid-length side = √3.



The rules

1. Anchor to the Angle
   Never label the adjacent or opposite sides until you have clearly marked the reference angle θ, as shifting the angle swaps those two labels completely.
   
   
2. Hypotenuse Never Changes
   Remember that the hypotenuse is locked down by geometry; it is always the longest side and always sits directly opposite the right angle.
   
   
3. Use Inverse to Extract Angles
   Use normal sin, cos, or tan when searching for a side length, but always use the inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) when solving for a missing angle.
   
   
4. Cosine is Sine Reversed
   When learning the exact value patterns, remember that the cosine values are simply the exact sine values running in the perfectly opposite order.
   

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Difficulty levels:

For the tables of facts, there are just a set of questions that do not change, and as such as with all other lists in this system the questions are unaffected by the difficulty level selected.
For the trig exact values option, the numbers get bigger with each level.