Trigonometry Worksheets

What it is

Trigonometric ratios relate an angle to side ratios in a right triangle. Special angles (0°, 30°, 45°, 60°, 90°) have exact values worth memorising. For any other angle, a calculator gives a decimal approximation. The sine law and cosine law extend these ideas to ANY triangle, not just right triangles, and the reciprocal ratios (csc, sec, cot) are just 1 divided by sin, cos, and tan.

Why it matters

Trig ratios underpin geometry, physics, waves, and the entire study of periodic functions in senior math; the sine and cosine laws extend that reach beyond right triangles to any triangle.

How to do it

1. Identify the ratio (sine, cosine, or tangent) and the angle.
2. If the angle is 0°, 30°, 45°, 60°, or 90°, recall the exact special-angle value from the unit circle, write it exactly (keep radicals, don't round).
3. For any other angle, use a calculator and round the decimal answer, these answers use ≈, not =, because they're approximations.
4. For an oblique (non-right) triangle: use the sine law when you know two angles and a side; use the cosine law when you know two sides and the included angle, or all three sides.

Examples by level

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

• Mixing up sine, cosine, and tangent (SOH-CAH-TOA).
• Rounding an exact special-angle value like √2/2 instead of leaving it exact.
• Not rounding calculator (≈) answers consistently.
• Treating tan 90° (or csc 0°, sec 90°, cot 0°) as a number, it's undefined.
• Using the sine law when two sides and the INCLUDED angle are given, that's a cosine-law setup.

Tips

• SOH-CAH-TOA: sine = opp/hyp, cosine = adj/hyp, tangent = opp/adj.
• tan 90° is undefined, and so is any reciprocal ratio whose base ratio is 0 (csc 0°, sec 90°, cot 0°).
• Answers with "=" are exact; answers with "≈" are calculator approximations.
• A compound angle like 15° isn't itself a special angle, but it's a sum or difference of two that are — rewrite it first, then apply the formula.

Key vocabulary

sine

opposite over hypotenuse

cosine

adjacent over hypotenuse

tangent

opposite over adjacent

sine law

a/sin(A) = b/sin(B) = c/sin(C), relates a triangle's sides to the sines of their opposite angles

cosine law

c² = a² + b² − 2ab·cos(C), a generalization of the Pythagorean theorem to any triangle

reciprocal ratio

csc, sec, and cot — 1 divided by sin, cos, and tan respectively

compound angle formula

a formula for sin/cos of a sum or difference of two angles, e.g. sin(A − B) = sinA cosB − cosA sinB

Frequently asked questions

What is SOH-CAH-TOA?

A memory aid: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

What is sin 30°?

Exactly 1/2, one of the special-angle values worth memorising.

Why do some answers have a ≈ instead of =?

Only the 5 special angles (0°, 30°, 45°, 60°, 90°) have exact values. Every other angle, and every sine-law or cosine-law answer, is evaluated with a calculator and rounded, so its answer is an approximation, marked with ≈.

When do I use the sine law vs. the cosine law?

Sine law when you know two angles and a side (or two sides and a non-included angle). Cosine law when you know two sides and the included angle, or all three sides.

What are csc, sec, and cot?

The reciprocal trig ratios: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.

How do you find the exact value of sin 15°?

15° isn't a special angle, but it's 45° − 30°, both of which are. Use the compound angle formula sin(A − B) = sinA cosB − cosA sinB with A=45° and B=30° to get sin 15° = (√6 − √2)/4.