--- 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, with the acute values 30, 45 and 60 taken from a halved equilateral triangle and a half square, and the endpoints 0 " 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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, with the acute values 30, 45 and 60 taken from a halved equilateral triangle and a half square, and the endpoints 0 and 90 read off the unit circle. 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 exact sin, cos and tan values for 0, 30, 45, 60 and 90 degrees 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 In a right angled triangle, pick one acute angle and call it theta. The hypotenuse is the longest side, opposite the right angle. The opposite side sits across from theta, and the adjacent side is the remaining leg. For that chosen angle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. The special angles are 0, 30, 45, 60, and 90 degrees. Here is the full table. sin: 0, 1/2, root 2 over 2, root 3 over 2, 1. cos: 1, root 3 over 2, root 2 over 2, 1/2, 0. tan: 0, root 3 over 3, 1, root 3, undefined. *(drawing: Sine, cosine, and tangent defined for angle theta.)* The 30 and 60 values come from a halved equilateral triangle. Cut an equilateral triangle of side 2 in half. The hypotenuse is the old side, so it is 2. The cut splits the base in half, so the shortest leg is 1. Pythagoras gives the long leg: long leg squared plus 1 squared equals 2 squared, so the long leg is root 3. Now read the ratios. For 30 degrees, opposite is 1 and hypotenuse is 2, so sin 30 = 1/2. For 60 degrees, opposite is root 3 and hypotenuse is 2, so sin 60 = root 3 over 2. Cosine uses the adjacent side, so cos 30 = root 3 over 2 and cos 60 = 1/2. Tangent is opposite over adjacent, so tan 30 = root 3 over 3 and tan 60 = root 3. Cut a square in half instead and you get a 45-45-90 triangle with equal legs of 1 and hypotenuse root 2, so sin 45 = cos 45 = root 2 over 2 and tan 45 = 1. Leave answers with roots in surd form, like root 3 over 2, because they never come out neat as decimals. *(drawing: At 0 degrees the point is (1, 0), at 90 degrees it is (0, 1).)* The right-triangle definitions need acute angles, so the endpoints 0 and 90 degrees need a small extension. Measure the angle from the positive x-axis and use a unit circle of radius 1 centred at the vertex. A ray at angle theta meets the circle at (cos theta, sin theta). At 0 degrees the point is (1, 0), so cos 0 = 1 and sin 0 = 0. At 90 degrees the point is (0, 1), so cos 90 = 0 and sin 90 = 1. Tangent is y over x where x is not zero, so tan 90 asks for 1 over 0, which is undefined. **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 - [Congruent Triangles and the Congruence Conditions](https://lightmysky.com/learn/mathematics/congruent-triangles-and-the-congruence-conditions-mt_it8t7mY71J) ## Optional connections - [Surds: Simplifying and Rationalising](https://lightmysky.com/learn/mathematics/surds-simplifying-and-rationalising-mt_cqr0crUUXS) ## 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)