E: Trigonometry: Understand and use the standard small angle approximations of sine, cosine and tangent sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ where θ is in radians.

The idea

For small angles measured in radians, the trigonometric functions behave almost like simple polynomials: sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − θ²/2. These match the leading terms of the Maclaurin series, and they are accurate only when θ is small and in radians — using degrees breaks them completely. A useful check is θ = 0.1: sin 0.1 ≈ 0.099833, close to 0.1; cos 0.1 ≈ 0.995004, close to 1 − 0.005 = 0.995. To use them, replace each function by its approximation, simplify algebraically, and keep the leading terms. They are especially powerful for limits such as (1 − cos θ)/θ² → 1/2 as θ → 0, and for estimating values without a calculator.

What earns the marks

• States that the approximations are valid only for small θ measured in radians, and explains that degree measure would give different numerical results.
• Quotes the three approximations accurately: sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ.
• Uses the approximations to simplify an expression or evaluate a limit, for example showing (1 − cos θ)/θ² tends to 1/2 as θ tends to 0.
• Recognises when a question expects an approximation rather than an exact value, and states the small-angle condition explicitly.
• Combines the approximations with standard trigonometric identities, such as sin 2θ ≈ 2θ or tan θ/sin θ ≈ 1 for small θ.
• Justifies the approximations by reference to the behaviour of the graphs near the origin or to the first terms of the series.

Common mistakes and how to put them right

• Applying the approximations when θ is not small, for example using sin 1.5 ≈ 1.5.

  Put it right: The error is large and the approximation is invalid, so check that θ is small before using it.

• Using degrees instead of radians, for example taking sin 30° ≈ 30.

  Put it right: The approximations require radian measure, so convert angles to radians first.

• Approximating cos θ by 1 alone when a θ² term matters, for example in (1 − cos θ)/θ².

  Put it right: Retaining 1 − θ²/2 gives the correct limiting behaviour, so keep the quadratic term when it affects the answer.

Exam tips

• Write down the approximation you are using before substituting, so the examiner can see the method even if arithmetic slips.
• For a limit such as (1 − cos θ)/θ², substitute the approximation cos θ ≈ 1 − θ²/2 first, then simplify; this is the expected method rather than dividing by the highest power of θ.
• Check the size of θ in the question; if it is not small, say so and use an exact or calculator method instead.
• Keep answers in exact surd or fraction form where the approximation produces them, rather than rounding prematurely.