Hard A-Level Small angle approximations Questions

Challenging, exam-style A-Level Small angle approximations questions with worked solutions. Stretch yourself on the hardest small-angle, simplify, cosine, product problems.

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A-Level34 questionsStep-by-step solutions
Question 1
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}
Question 2
8 markschallenging
Using small-angle approximations, which is the best approximation to sin⁡(4x)tan⁡(x)x2\frac{\sin{\left(4 x \right)} \tan{\left(x \right)}}{x^{2}} for small xx?
Show worked solution

Worked solution

  1. Write the expression to be approximated

    sin⁡(4x)tan⁡(x)x2\frac{\sin{\left(4 x \right)} \tan{\left(x \right)}}{x^{2}}

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

  2. Substitute the small-angle approximations

    4x2x2\frac{4 x^{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

    44

    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

    sin⁡(4x)tan⁡(x)x2≈4\frac{\sin{\left(4 x \right)} \tan{\left(x \right)}}{x^{2}}\approx 4

    This matches the small-angle approximation.

Answer
44
Question 3
8 markschallenging
Using small-angle approximations, which is the best approximation to tan⁡(6x)sin⁡(3x)\frac{\tan{\left(6 x \right)}}{\sin{\left(3 x \right)}} for small xx?
Show worked solution

Worked solution

  1. Write the expression to be approximated

    tan⁡(6x)sin⁡(3x)\frac{\tan{\left(6 x \right)}}{\sin{\left(3 x \right)}}

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

  2. Substitute the small-angle approximations

    6x3x\frac{6 x}{3 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. Simplify the result

    22

    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

    tan⁡(6x)sin⁡(3x)≈2\frac{\tan{\left(6 x \right)}}{\sin{\left(3 x \right)}}\approx 2

    This matches the small-angle approximation.

Answer
22
Question 4
8 markschallenging
Which of the following statements about small-angle approximations is correct?
Show worked solution

Worked solution

  1. Recall the relevant approximation

    tan⁡x>x for small x>0\tan x> x\ \text{for small }x>0

    This is the mathematics behind the correct statement.

  2. Assess each statement in turn

    compare every option with the approximation\text{compare every option with the approximation}

    Only one option is consistent with the approximation.

  3. 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.

  4. 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.

  5. 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).

  6. 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.

  7. 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.

  8. Interpret the behaviour as a limit

    x→0x\to 0

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

  9. Compare sine and tangent

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

    Both share the same first-order approximation.

  10. 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.

  11. 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.

  12. Confirm radians are being used

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

    Degrees must be converted to radians first.

  13. Write cosine in its quadratic form

    cos⁡x≈1−x22\cos x\approx 1-\tfrac{x^{2}}{2}

    This is the second-order approximation for cosine.

  14. Write the linear approximations

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

    These are the first-order approximations.

  15. Identify the correct statement

    tan⁡x>x for small x>0\tan x> x\ \text{for small }x>0

    The correct statement follows directly from this approximation.

Answer
The approximation tan(x) is approximately x underestimates tan(x) for small positive x.
Question 5
8 markschallenging
Which of the following statements about small-angle approximations is correct?
Show worked solution

Worked solution

  1. Recall the relevant approximation

    θ∘=θ⋅π180 rad\theta^{\circ}=\theta\cdot\tfrac{\pi}{180}\text{ rad}

    This is the mathematics behind the correct statement.

  2. Assess each statement in turn

    compare every option with the approximation\text{compare every option with the approximation}

    Only one option is consistent with the approximation.

  3. 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.

  4. 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.

  5. 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).

  6. 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.

  7. 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.

  8. Interpret the behaviour as a limit

    x→0x\to 0

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

  9. Compare sine and tangent

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

    Both share the same first-order approximation.

  10. 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.

  11. 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.

  12. Confirm radians are being used

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

    Degrees must be converted to radians first.

  13. Write cosine in its quadratic form

    cos⁡x≈1−x22\cos x\approx 1-\tfrac{x^{2}}{2}

    This is the second-order approximation for cosine.

  14. Write the linear approximations

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

    These are the first-order approximations.

  15. Identify the correct statement

    θ∘=θ⋅π180 rad\theta^{\circ}=\theta\cdot\tfrac{\pi}{180}\text{ rad}

    The correct statement follows directly from this approximation.

Answer
To use the approximations, an angle given in degrees must first be converted to radians.

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