A-level · Beta

Small-angle approximations

Compare exact trigonometric values with approximations and measure the error.

\sinx≈x,x=10π180≈0.17453\sinx\approx x,\quad x=10\frac{\pi}{180}\approx 0.17453
Small-angle approximations-0.8-1.1-0.4-0.55000.40.550.81.1Angle x (radians)sin x

Scroll the diagram sideways to see every label.

Angle in radians
0.17453
Exact trig value (rounded)
0.17365
Approximation
0.17453
Absolute error
0.00088
Relative error
0.50951%

At 10°, the sin approximation has about 0.50951% relative error.

How is the error measured?

absolute error=∣approximation−exact value∣\text{absolute error}=|\text{approximation}-\text{exact value}|

relative error=absolute error∣exact value∣\text{relative error}=\frac{\text{absolute error}}{|\text{exact value}|}

Multiply relative error by 100 for a percentage. A useful angle range depends on the accuracy you need. The same cutoff need not suit all three approximations.

Blue: exact function; red dashed: approximation. The graph uses radians. Degree selections are converted before calculating; substituting the degree number directly is incorrect. The graph shows a limited interval near zero, and displayed values are rounded to 5 decimal places.

Predict, test and explain

Step 1 of 3

Predict

To use sin x ≈ x for an angle of 5°, can you substitute x = 5?

InvestigateWhy must you convert degrees to radians?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

To use sin x ≈ x for an angle of 5°, can you substitute x = 5?

sin⁡ ⁣(π18)≈π18\sin\!\left(\frac{\pi}{18}\right)\approx\frac{\pi}{18}

Hint

Convert degrees to radians using x = degrees × π/180.

Reveal answer

The standard forms sin x ≈ x, tan x ≈ x and cos x ≈ 1 − x²/2 require radians. Their accuracy improves near zero, and the acceptable error depends on the calculation.

∣π18−sin⁡ ⁣(π18)∣≈0.000885\left|\frac{\pi}{18}-\sin\!\left(\frac{\pi}{18}\right)\right|\approx0.000885

Build an explanation

Hint 1

Convert degrees to radians using x = degrees × π/180.

Hint 2

Apply the matching approximation: x for sine or tangent, and 1 − x²/2 for cosine.

Hint 3

Compare the approximation with the reference value using absolute error. A small angle alone does not specify the accuracy required.

Worked example

This example uses fixed values, separate from the diagram controls.

Use a small-angle approximation for sin 10° and calculate its absolute error.

  1. Convert 10° to π/18 radians, approximately 0.174533.
  2. Use sin x ≈ x, giving sin 10° ≈ 0.174533.
  3. The reference value is approximately 0.173648, so the absolute error is about |0.174533 − 0.173648| = 0.000885. Use unrounded values when calculating.

Answer: sin 10° ≈ π/18, with absolute error approximately 0.000885.

Connect this idea

Watch out: The standard forms sin x ≈ x, tan x ≈ x and cos x ≈ 1 − x²/2 require radians. Their accuracy improves near zero, and the acceptable error depends on the calculation. Convert the displayed degrees to radians before substituting. These approximations are local to zero, and acceptable error depends on the task. Absolute error is the magnitude of approximate minus reference value. Displayed values are rounded; equality after rounding does not establish exact equality.

Specification and learning route

AQA E2 · Edexcel Pure 5.2

A-level. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Radians; Sine, cosine and tangent; Absolute error.

AQA specification · Edexcel specification