By Allison Boley
Updated Aug 30, 2022

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Trigonometry is the study of angles and their associated ratios—sine, cosine, and tangent. While modern calculators make these calculations instant, many exams and homework assignments require you to solve them manually. Below is a concise guide that covers all the essential methods, from memorizing key angles to using right‑triangle geometry and even classic trigonometric tables.

Trig Functions on the Coordinate Axes

Angles that align with the axes (0°, 90°, 180°, 270°) have simple, memorisable sine and cosine values:

• sin 0° = 0     cos 0° = 1
• sin 90° = 1     cos 90° = 0
• sin 180° = 0    cos 180° = –1
• sin 270° = –1   cos 270° = 0

These base cases serve as anchors for all other angles, especially when combined with symmetry arguments.

Right Triangles and the SOHCAHTOA Rule

When a problem presents a right triangle, the acronym SOHCAHTOA guides you through the trigonometric ratios:

• Sine = Opposite / Hypotenuse
• Cosine = Adjacent / Hypotenuse
• Tangent = Opposite / Adjacent

For example, consider a triangle with angles 90°, 12°, 78°. If the hypotenuse is 24 units and the side opposite the 12° angle is 5 units, then:

sin 12° = 5 / 24 ≈ 0.2083

Knowing the remaining side (adjacent) allows you to compute cosine and tangent in the same manner.

Special Right Triangles

Two classic right‑triangle configurations simplify many calculations:

• 30°–60°–90° triangle: side ratios 1 : √3 : 2 (opposite 30°, opposite 60°, hypotenuse)
• 45°–45°–90° triangle: side ratios 1 : 1 : √2 (isosceles right triangle)

These ratios let you instantly write down sine, cosine, and tangent for 30°, 60°, and 45°:

• sin 30° = 1 / 2  cos 30° = √3 / 2  tan 30° = 1 / √3
• sin 60° = √3 / 2  cos 60° = 1 / 2  tan 60° = √3
• sin 45° = √2 / 2  cos 45° = √2 / 2  tan 45° = 1

These base angles cover most “special” cases you’ll encounter in textbooks and exams.

Using Trigonometric Tables

When neither a right triangle nor a special angle is provided, trigonometric tables are a reliable fallback. They list pre‑computed values for sine, cosine, and tangent for every degree from 0° to 90° (and sometimes beyond). While digital tools are common, a paper table remains a valuable resource for manual calculation.

Resources

For a printable trig table, see the PDF in the Resources section of this article.