Trigonometry
Calculators, worked examples and guides — from SOH-CAH-TOA and right-angled triangles through to the sine rule, cosine rule and exact trigonometric values.
Learning about Trigonometry
The key idea: in any right-angled triangle, the ratios between the sides are fixed for a given angle — always. This is the remarkable fact that makes trigonometry work. SOH-CAH-TOA names those three ratios.
Type 1 — Finding a missing side
"Angle = 30°, hypotenuse = 10cm. Find the opposite side."
Label the sides: O = ?, A = adjacent, H = 10cm.
Choose ratio: sin (uses O and H) → sin(30°) = O/H.
Step 1: O = H × sin(30°).
Step 2: O = 10 × 0.5 = 5cm
Shortcut: SOH — label O and H first, then Sin = O/H rearranges to O = H × sin(θ)
Type 2 — Finding a missing angle
"Opposite = 6cm, adjacent = 8cm. Find the angle."
Choose ratio: tan (uses O and A) → tan(θ) = O/A.
Step 1: tan(θ) = 6/8 = 0.75.
Step 2: θ = tan⁻¹(0.75) ≈ 36.9°
Shortcut: Finding a SIDE uses sin/cos/tan. Finding an ANGLE uses sin⁻¹/cos⁻¹/tan⁻¹
Type 3 — Sine rule
"A triangle has angle A = 40°, angle B = 75°, and side a = 8cm. Find side b."
Formula: a/sin(A) = b/sin(B).
Step 1: 8/sin(40°) = b/sin(75°).
Step 2: b = 8 × sin(75°)/sin(40°).
Step 3: b ≈ 8 × 0.966/0.643 ≈ 12.0cm
Shortcut: Use the sine rule when you have a matched angle-side pair plus one other measurement
Trigonometry Formula Sheet & Rules
Need a printable revision sheet? Access SOHCAHTOA right-triangle rules, Sine Rule, Cosine Rule, non-right triangle area, exact values (30°, 45°, 60°), and key trig identities with step-by-step worked examples and custom print export.
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Trigonometry Questions Answered
What is trigonometry?
Trigonometry is the study of relationships between the angles and side lengths of triangles. It is used to find unknown sides or angles in right-angled triangles, and can be extended to any triangle using the sine and cosine rules.
What does SOH-CAH-TOA mean?
It is a mnemonic for the three trigonometric ratios in a right-angled triangle: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
What is the hypotenuse?
The hypotenuse is the longest side of a right-angled triangle — always the side directly opposite the right angle.
How do I find a missing angle using trigonometry?
Calculate the ratio of the two known sides, then apply the inverse function. For example, if tan(θ) = 0.75, then θ = tan⁻¹(0.75) ≈ 36.9°.
What is the sine rule?
The sine rule states that a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are side lengths and A, B, C are the angles opposite those sides. It allows you to solve any triangle when you know two angles and a side, or two sides and a non-included angle.