A-Level Small angle approximations Practice Questions

Free A-Level Small angle approximations practice questions with full step-by-step worked solutions. Covers small-angle, simplify, cosine, polynomial. Practise exam-style problems and check your method.

small-anglesimplifycosinepolynomialestimatevalidity
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Use small-angle approximations to simplify sin⁡(3x)x\frac{\sin{\left(3 x \right)}}{x} for small xx.
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Worked solution

  1. Write the expression to be approximated

    sin⁡(3x)x\frac{\sin{\left(3 x \right)}}{x}

    We approximate this expression for small xx (in radians).

  2. Substitute the small-angle approximations

    3xx\frac{3 x}{x}

    Replace sin⁡x\sin x and tan⁡x\tan x by xx, and cos⁡x\cos x by 1−x221-\tfrac{x^{2}}{2}.

  3. State the approximation

    sin⁡(3x)x≈3\frac{\sin{\left(3 x \right)}}{x}\approx 3

    This is the required small-angle approximation.

Answer
33
Question 2
2 markseasy
Find the percentage error when a small-angle approximation is used to estimate tan⁡(0.1)\tan(0.1). Give your answer to 3 significant figures.
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Worked solution

  1. Write the approximate and true values

    approx=0.1, true=0.100335\text{approx}=0.1,\ \text{true}=0.100335

    Compare the small-angle estimate with the calculator value.

  2. Write the percentage-error formula

    ∣approx−truetrue∣×100%\left|\frac{\text{approx}-\text{true}}{\text{true}}\right|\times 100\%

    Percentage error is measured relative to the true value.

  3. State the percentage error

    0.334%0.334\%

    This is the percentage error to 3 significant figures.

Answer
0.334%0.334\%
Question 3
3 marksintermediate
Using small-angle approximations, which is the best approximation to xtan⁡(x)+cos⁡(x)x \tan{\left(x \right)} + \cos{\left(x \right)} for small xx?
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Worked solution

  1. Write the expression to be approximated

    xtan⁡(x)+cos⁡(x)x \tan{\left(x \right)} + \cos{\left(x \right)}

    We approximate this expression for small xx (in radians).

  2. Substitute the small-angle approximations

    x2+1−x22x^{2} + 1 - \frac{x^{2}}{2}

    Replace sin⁡x\sin x and tan⁡x\tan x by xx, and cos⁡x\cos x by 1−x221-\tfrac{x^{2}}{2}.

  3. Simplify the result

    x22+1\frac{x^{2}}{2} + 1

    Cancel common factors and discard higher-order terms.

  4. Recall the approximations used

    sin⁡x→x, tan⁡x→x, cos⁡x→1−x22\sin x\to x,\ \tan x\to x,\ \cos x\to 1-\tfrac{x^{2}}{2}

    These are the standard small-angle results.

  5. Recall the standard small-angle approximations

    sin⁡x≈x,tan⁡x≈x,cos⁡x≈1−x22\sin x\approx x,\quad \tan x\approx x,\quad \cos x\approx 1-\tfrac{x^{2}}{2}

    These are the three results used throughout.

  6. Select the correct approximation

    xtan⁡(x)+cos⁡(x)≈x22+1x \tan{\left(x \right)} + \cos{\left(x \right)}\approx \frac{x^{2}}{2} + 1

    This matches the small-angle approximation.

Answer
x22+1\frac{x^{2}}{2} + 1
Question 4
5 markshard
Using small-angle approximations, which is the best approximation to sin⁡(2x)+tan⁡(3x)\sin{\left(2 x \right)} + \tan{\left(3 x \right)} for small xx?
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Worked solution

  1. Write the expression to be approximated

    sin⁡(2x)+tan⁡(3x)\sin{\left(2 x \right)} + \tan{\left(3 x \right)}

    We approximate this expression for small xx (in radians).

  2. Substitute the small-angle approximations

    5x5 x

    Replace sin⁡x\sin x and tan⁡x\tan x by xx, and cos⁡x\cos x by 1−x221-\tfrac{x^{2}}{2}.

  3. Confirm the simplest form

    sin⁡(2x)+tan⁡(3x)≈5x\sin{\left(2 x \right)} + \tan{\left(3 x \right)}\approx 5 x

    The substituted expression is already fully simplified.

  4. Recall the approximations used

    sin⁡x→x, tan⁡x→x, cos⁡x→1−x22\sin x\to x,\ \tan x\to x,\ \cos x\to 1-\tfrac{x^{2}}{2}

    These are the standard small-angle results.

  5. Recall the standard small-angle approximations

    sin⁡x≈x,tan⁡x≈x,cos⁡x≈1−x22\sin x\approx x,\quad \tan x\approx x,\quad \cos x\approx 1-\tfrac{x^{2}}{2}

    These are the three results used throughout.

  6. State the validity condition

    x small and measured in radiansx\text{ small and measured in radians}

    The approximations only hold for small angles in radians.

  7. Note where the approximations come from

    from the Maclaurin series of sin⁡, cos⁡ and tan⁡\text{from the Maclaurin series of }\sin,\ \cos\ \text{and}\ \tan

    Each series is truncated after the leading term(s).

  8. Explain why cosine keeps a quadratic term

    cos⁡x=1−x22+⋯(no linear term)\cos x = 1-\tfrac{x^{2}}{2}+\cdots\quad(\text{no linear term})

    The linear term of cosine is zero, so the quadratic term is retained.

  9. Identify the neglected terms

    terms of order x3 and higher are discarded\text{terms of order }x^{3}\text{ and higher are discarded}

    Higher powers of a small number are negligible.

  10. Select the correct approximation

    sin⁡(2x)+tan⁡(3x)≈5x\sin{\left(2 x \right)} + \tan{\left(3 x \right)}\approx 5 x

    This matches the small-angle approximation.

Answer
5x5 x
Question 5
8 markschallenging
Using small-angle approximations, which is the best approximation to x2+cos⁡(2x)x^{2} + \cos{\left(2 x \right)} for small xx?
Show worked solution

Worked solution

  1. Write the expression to be approximated

    x2+cos⁡(2x)x^{2} + \cos{\left(2 x \right)}

    We approximate this expression for small xx (in radians).

  2. Substitute the small-angle approximations

    x2+1−2x2x^{2} + 1 - 2 x^{2}

    Replace sin⁡x\sin x and tan⁡x\tan x by xx, and cos⁡x\cos x by 1−x221-\tfrac{x^{2}}{2}.

  3. Simplify the result

    1−x21 - x^{2}

    Cancel common factors and discard higher-order terms.

  4. Recall the approximations used

    sin⁡x→x, tan⁡x→x, cos⁡x→1−x22\sin x\to x,\ \tan x\to x,\ \cos x\to 1-\tfrac{x^{2}}{2}

    These are the standard small-angle results.

  5. Recall the standard small-angle approximations

    sin⁡x≈x,tan⁡x≈x,cos⁡x≈1−x22\sin x\approx x,\quad \tan x\approx x,\quad \cos x\approx 1-\tfrac{x^{2}}{2}

    These are the three results used throughout.

  6. State the validity condition

    x small and measured in radiansx\text{ small and measured in radians}

    The approximations only hold for small angles in radians.

  7. Note where the approximations come from

    from the Maclaurin series of sin⁡, cos⁡ and tan⁡\text{from the Maclaurin series of }\sin,\ \cos\ \text{and}\ \tan

    Each series is truncated after the leading term(s).

  8. Explain why cosine keeps a quadratic term

    cos⁡x=1−x22+⋯(no linear term)\cos x = 1-\tfrac{x^{2}}{2}+\cdots\quad(\text{no linear term})

    The linear term of cosine is zero, so the quadratic term is retained.

  9. Identify the neglected terms

    terms of order x3 and higher are discarded\text{terms of order }x^{3}\text{ and higher are discarded}

    Higher powers of a small number are negligible.

  10. Interpret the behaviour as a limit

    x→0x\to 0

    The approximation is exact in the limit as the angle tends to zero.

  11. Compare sine and tangent

    sin⁡x≈x≈tan⁡x\sin x\approx x\approx \tan x

    Both share the same first-order approximation.

  12. Keep terms to the required order

    retain terms up to x2\text{retain terms up to }x^{2}

    We only need accuracy to second order here.

  13. Note that the error shrinks with the angle

    error→0 as x→0\text{error}\to 0\ \text{as}\ x\to 0

    Smaller angles give more accurate approximations.

  14. Confirm radians are being used

    1 rad=180π∘1\text{ rad}=\tfrac{180}{\pi}^{\circ}

    Degrees must be converted to radians first.

  15. Select the correct approximation

    x2+cos⁡(2x)≈1−x2x^{2} + \cos{\left(2 x \right)}\approx 1 - x^{2}

    This matches the small-angle approximation.

Answer
1−x21 - x^{2}

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