---
title: "Exact Values of Sine, Cosine and Tangent"
description: "Know and use the exact values of sin, cos and tan for 0, 30, 45, 60 and 90 degrees, taken from a halved equilateral triangle and a half square."
canonical: https://lightmysky.com/learn/mathematics/exact-values-of-sine-cosine-and-tangent-mt_SNJEbXYOrR
source: https://lightmysky.com/learn/mathematics/exact-values-of-sine-cosine-and-tangent-mt_SNJEbXYOrR.md
retrieved: 2026-09-12
---

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# Exact Values of Sine, Cosine and Tangent

Know and use the exact values of sin, cos and tan for 0, 30, 45, 60 and 90 degrees, taken from a halved equilateral triangle and a half square.

Subject: Mathematics · Area: Geometry · Ages 15 to 16
Page: https://lightmysky.com/learn/mathematics/exact-values-of-sine-cosine-and-tangent-mt_SNJEbXYOrR

## Ready when they can

- Recall sin 30, cos 60 and tan 45 without a calculator
- Derive the 30 and 60 degree values from an equilateral triangle cut in half
- Leave an answer in surd form instead of rounding it

## Lesson: Angles you just know

Some angles appear so often that you must know their sine, cosine, and tangent with no calculator. Your must-know trio is sin 30 = 1/2, cos 60 = 1/2, and tan 45 = 1. A close friend is sin 60 = root 3 over 2, which you leave in surd form.

Each value comes from a special triangle, so learn the source and you will never mix them up. Cut an equilateral triangle in half and you get a 30-60-90 triangle: the hypotenuse is the old side, the shortest leg is half of it, and that half-over-whole is exactly sin 30 = 1/2. Cut a square in half instead and you get a 45-45-90 triangle with equal legs, so opposite over adjacent gives tan 45 = 1.

**Tip.** Practise by covering the table, writing what you remember, then checking your gaps. Leave an answer in surd form whenever it holds a root that never comes out neat, like root 3 over 2. And use the symmetries to check yourself: sin 30 matches cos 60 because 30 and 60 are co-angles.

**Recap.** Halve an equilateral triangle for 30 and 60, halve a square for 45, and keep roots in surd form.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [Surds: Simplifying and Rationalising](https://lightmysky.com/learn/mathematics/surds-simplifying-and-rationalising-mt_cqr0crUUXS)
- [Congruent Triangles and the Congruence Conditions](https://lightmysky.com/learn/mathematics/congruent-triangles-and-the-congruence-conditions-mt_it8t7mY71J)

## Opens up

- [The Sine Rule](https://lightmysky.com/learn/mathematics/the-sine-rule-mt_cTt_-9KZho)
- [Projectiles Launched at an Angle](https://lightmysky.com/learn/science/projectiles-launched-at-an-angle-mt_MMuacyCqb8)
- [Radian Measure and Sectors](https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4)