--- title: "The Unit Circle and the Graphs of Sine, Cosine and Tangent" description: "Define sine and cosine as the coordinates of a point travelling round the unit circle, which extends them beyond a right angle and explains the period, symmetry and repeated values of their graphs." canonical: https://lightmysky.com/learn/mathematics/the-unit-circle-and-the-graphs-of-sine-cosine-and-tangent-mt_VJaimevi3- source: https://lightmysky.com/learn/mathematics/the-unit-circle-and-the-graphs-of-sine-cosine-and-tangent-mt_VJaimevi3-.md retrieved: 2026-09-12 --- > **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`. # The Unit Circle and the Graphs of Sine, Cosine and Tangent Define sine and cosine as the coordinates of a point travelling round the unit circle, which extends them beyond a right angle and explains the period, symmetry and repeated values of their graphs. Subject: Mathematics · Area: Geometry · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/the-unit-circle-and-the-graphs-of-sine-cosine-and-tangent-mt_VJaimevi3- ## Ready when they can - Sketch y = sin x from 0 to 2π and mark where it repeats - Use the circle to explain why sin(π - x) = sin x - State the period and amplitude of y = 2 sin 3x and say why tan has asymptotes - Read sine and cosine from unit circle coordinates, for example sin 30 is 0.5, cos 60 is 0.5 and sin 90 is 1. - Use the circle to explain why cosine turns negative in the second quadrant, for example cos 120 is -0.5 via reference 60. ## Lesson: One circle that draws every wave Picture a bike wheel spinning, with a dot painted on the rim. As the wheel turns, the dot rises and falls, first fast, then slow. The unit circle captures that same rise and fall with radius 1. Each angle from the positive x axis lands the dot on a point: read x for cosine and y for sine. At 90 degrees the point is (0, 1), so sine is 1 and cosine is 0. Let the angle grow from 0 to 2 pi and track the height. That trace is the graph of y equals sin x. It repeats every full turn, so sine and cosine have period 2 pi. That repeat is why a held musical note sounds steady, so mark where the repeat starts. Symmetry gives mirror rules. Angles x and 180 degrees minus x share the same height, so sin of pi minus x equals sin x. Their x coordinates flip sign. In the second quadrant sine stays positive while cosine turns negative. **Example.** Take y equals 2 sin 3x. The outer 2 sets the amplitude, so the wave is twice as tall. The inner 3 squeezes three waves into one turn, so the period is 2 pi over 3. Tangent breaks where cosine is zero, at 90 and 270 degrees, and climbs between. **Recap.** Coordinates give the values, the travelling point draws the graph, and symmetry explains the repeats. ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Transformations of Graphs](https://lightmysky.com/learn/mathematics/transformations-of-graphs-mt_691_c-7Z5M) - [Radian Measure and Sectors](https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4) ## Opens up - [Solving Trigonometric Equations in a Given Interval](https://lightmysky.com/learn/mathematics/solving-trigonometric-equations-in-a-given-interval-mt_74_jiK2xXA) - [Small Angle Approximations](https://lightmysky.com/learn/mathematics/small-angle-approximations-mt_fIsZes5Os3) - [The Pythagorean and Quotient Identities](https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh) - [Polar Coordinates and Polar Curves](https://lightmysky.com/learn/mathematics/polar-coordinates-and-polar-curves-mt_R5QMUJvA1v)