--- title: "Small Angle Approximations" description: "Near zero, and only in radians, the sine of an angle is the angle, the tangent of it is the angle too, and the cosine drops off as one minus half the square. These are the approximations that turn an " canonical: https://lightmysky.com/learn/mathematics/small-angle-approximations-mt_fIsZes5Os3 source: https://lightmysky.com/learn/mathematics/small-angle-approximations-mt_fIsZes5Os3.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`. # Small Angle Approximations Near zero, and only in radians, the sine of an angle is the angle, the tangent of it is the angle too, and the cosine drops off as one minus half the square. These are the approximations that turn an equation nobody can solve into one anybody can, and knowing when they stop being safe is half the point. Subject: Mathematics · Area: Geometry · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/small-angle-approximations-mt_fIsZes5Os3 ## Ready when they can - States the three approximations, insists on radians, and says what goes wrong in degrees - Reads the approximations off the graphs of sine, cosine and tangent near the origin - Uses an approximation to simplify an expression, then checks the size of the error it introduced at a stated angle ## Lesson: Tiny angles made simple Near zero, and only in radians, three shortcuts hold: sin x behaves like x, tan x behaves like x too, and cos x behaves like 1 minus x squared over 2. Radians are non negotiable because arc length equals radius times angle only in radians. Degree based slopes would drag pi over 180 factors into every derivative, so convert to radians first and approximate second. The graphs show why the rules work. Sine hugs the diagonal y equals x near the origin, since both share the same value and the same slope there. Cosine peaks flat at 1 and falls off quadratically, so 1 minus cos x behaves like x squared over 2. Since sin behaves like x and cos behaves like 1, their ratio tan behaves like x. **Example.** Estimate cos 0.2 as 1 minus 0.04 over 2, which is 0.98. Estimate cos 0.1 as 1 minus 0.01 over 2, which is 0.995. Replace sin 0.06 with 0.06 and the fraction sin 0.06 over 0.06 comes out about 1. After replacing sin 0.3 with 0.3, the next Taylor term x cubed over 6 prices the mistake at 0.0045. **Tip.** Always quote the value with its price. The dropped terms measure the damage: x cubed over 6 bounds sine error and x to the fourth over 24 bounds cosine error. Halving x shrinks the sine error bound eightfold, so smaller angles are dramatically safer. **Recap.** In radians, sine and tangent mimic x while cosine dips quadratically, each with a priced error. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Radian Measure and Sectors](https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4) - [The Unit Circle and the Graphs of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/the-unit-circle-and-the-graphs-of-sine-cosine-and-tangent-mt_VJaimevi3-) ## Opens up - [Derivatives of Exponential, Logarithmic and Trigonometric Functions](https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT) - [Simple Harmonic Motion: the Defining Relation and Its Graphs](https://lightmysky.com/learn/science/simple-harmonic-motion-the-defining-relation-and-its-graphs-mt_NTDayHkdfh)