Mathematics · June 2026

Trigonometry Basics Explained

Trigonometry is the study of the relationships between the angles and side lengths of triangles. It sounds narrow, but it underpins navigation, engineering, physics, music, and computer graphics. Once you understand the three core ratios, most of the rest follows naturally.

The Right Triangle and SOH-CAH-TOA

Start with a right-angled triangle — one angle is exactly 90 degrees. Label one of the other angles θ (theta). The three sides then have specific names relative to θ:

The hypotenuse is the longest side, always opposite the right angle.
The opposite side is the side directly across the triangle from θ.
The adjacent side is the side next to θ that is not the hypotenuse.

The three primary trigonometric ratios are defined as fractions of these sides:

Sine: sin(θ) = opposite / hypotenuse
Cosine: cos(θ) = adjacent / hypotenuse
Tangent: tan(θ) = opposite / adjacent

The mnemonic SOH-CAH-TOA is almost universally used to remember these: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Worked example: a right triangle has an angle of 35° and a hypotenuse of 10 cm. To find the opposite side: sin(35°) = opposite / 10, so opposite = 10 × sin(35°) ≈ 10 × 0.574 = 5.74 cm.

Finding Angles from Sides: Inverse Trig Functions

If you know two sides and want the angle, use the inverse trigonometric functions: sin⁻¹, cos⁻¹, tan⁻¹ (also written arcsin, arccos, arctan).

Example: a right triangle has an opposite side of 4 cm and a hypotenuse of 7 cm. Then sin(θ) = 4/7 ≈ 0.571, so θ = sin⁻¹(0.571) ≈ 34.8°.

One frequent error: students confuse sin⁻¹(x) with 1/sin(x). They are completely different. The inverse function undoes the sine operation to recover the angle; the reciprocal (1/sin) is a different function called cosecant.

Exact Values You Must Know

For certain "special" angles, the trig ratios produce exact values — no calculator needed. These come from two standard right triangles:

The 45-45-90 triangle (isosceles right triangle with legs of length 1): hypotenuse = √2. This gives sin(45°) = cos(45°) = 1/√2 = √2/2, and tan(45°) = 1.

The 30-60-90 triangle (half of an equilateral triangle with side 2): the short leg is 1, the long leg is √3, hypotenuse is 2. This gives: sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3; sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3.

These exact values appear repeatedly in exams. Build a small table and memorise it.

Radians: The Natural Angle Unit

Degrees are a human convention (360 chosen for its divisibility). Mathematics and physics prefer radians, defined as the arc length subtended on a unit circle. A full circle has a circumference of 2π, so 360° = 2π radians. The conversion factor is: degrees × π/180 = radians; radians × 180/π = degrees.

Key conversions: 0° = 0, 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π. For most advanced mathematics and all calculus, angles must be in radians — the derivative of sin(x) is cos(x) only when x is measured in radians.

The Unit Circle and Extending Beyond 90 Degrees

The unit circle is a circle of radius 1 centred at the origin of a coordinate plane. It extends the meaning of sin and cos beyond the 0°–90° range of a right triangle to any angle.

For any angle θ measured anticlockwise from the positive x-axis, the point where the terminal ray meets the unit circle has coordinates (cosθ, sinθ). This definition makes sin and cos periodic functions that repeat every 360° (or 2π radians) and explains how they can take both positive and negative values.

The CAST rule (or a unit-circle diagram) tells you which ratios are positive in each quadrant: All positive in quadrant I, Sine positive in quadrant II, Tangent positive in quadrant III, Cosine positive in quadrant IV.

For any angle in quadrant II, III, or IV, you find the trig ratio by first finding the reference angle (the acute angle between the terminal ray and the x-axis), applying the ratio to the reference angle, and then attaching the correct sign using the CAST rule. For example, sin(150°): reference angle is 30°, sine is positive in quadrant II, so sin(150°) = sin(30°) = 1/2.

The Pythagorean Identity

The most important trigonometric identity follows directly from the Pythagorean theorem. On the unit circle, the point (cosθ, sinθ) lies on a circle of radius 1, so by Pythagoras: cos²(θ) + sin²(θ) = 1. This identity holds for every angle and is used constantly to simplify expressions, prove other identities, and solve equations. Two related identities: 1 + tan²(θ) = sec²(θ) and 1 + cot²(θ) = csc²(θ), both derived by dividing the main identity by cos² or sin² respectively.

Solving Non-Right Triangles: Sine and Cosine Rules

SOH-CAH-TOA only applies directly to right triangles. For triangles with no right angle, two more powerful tools are needed.

The sine rule: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the side lengths and A, B, C are the angles opposite those sides. Use the sine rule when you know: two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA — but beware of the ambiguous case where two triangles may satisfy the given information).

The cosine rule: a² = b² + c² − 2bc cos(A). Use it when you know: two sides and the included angle (SAS), or all three sides (SSS) and want to find an angle. To find angle A from three known sides, rearrange: cos(A) = (b² + c² − a²) / (2bc). Notice that when A = 90°, cos(A) = 0 and the cosine rule reduces to Pythagoras's theorem — a reassuring consistency check.

Common Applications

Angles of elevation and depression are measured from the horizontal. If you stand 50 m from a tower and look up to its top at an elevation angle of 28°, the height is 50 × tan(28°) ≈ 26.6 m. If you look down from a cliff at an angle of depression of 15° to a boat, the geometry is the same — the angle of depression equals the alternate angle of elevation.

In physics, trig resolves forces and velocities into components. A force of 40 N acting at 30° above horizontal has a horizontal component of 40 cos(30°) ≈ 34.6 N and a vertical component of 40 sin(30°) = 20 N.

Summary

Trigonometry rests on three ratios in a right triangle: sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent (opposite/adjacent), remembered with SOH-CAH-TOA. Inverse functions recover angles from ratios. Exact values at 30°, 45°, and 60° should be memorised. Radians are the standard angle unit in advanced mathematics and calculus (360° = 2π). The unit circle extends the three ratios to any angle; the Pythagorean identity (sin²θ + cos²θ = 1) ties them together. For non-right triangles, the sine rule handles two angles and a side; the cosine rule handles two sides and their included angle, or all three sides.