Small angle approximations Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Small angle approximations questions. See exactly how to solve problems on small-angle, simplify, cosine, polynomial.

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.

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
Use small-angle approximations to simplify sin⁡(2x)x\frac{\sin{\left(2 x \right)}}{x} for small xx.

Worked solution

  1. Write the expression to be approximated

    sin⁡(2x)x\frac{\sin{\left(2 x \right)}}{x}

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

  2. Substitute the small-angle approximations

    2xx\frac{2 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⁡(2x)x≈2\frac{\sin{\left(2 x \right)}}{x}\approx 2

    This is the required small-angle approximation.

Answer
22
Question 3
2 markseasy
Use small-angle approximations to simplify 1−cos⁡(x)x2\frac{1 - \cos{\left(x \right)}}{x^{2}} for small xx.

Worked solution

  1. Write the expression to be approximated

    1−cos⁡(x)x2\frac{1 - \cos{\left(x \right)}}{x^{2}}

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

  2. Substitute the small-angle approximations

    x22x2\frac{\frac{x^{2}}{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. State the approximation

    1−cos⁡(x)x2≈12\frac{1 - \cos{\left(x \right)}}{x^{2}}\approx \frac{1}{2}

    This is the required small-angle approximation.

Answer
12\frac{1}{2}
Question 4
2 markseasy
Use small-angle approximations to simplify 1−cos⁡(x)xsin⁡(x)\frac{1 - \cos{\left(x \right)}}{x \sin{\left(x \right)}} for small xx.

Worked solution

  1. Write the expression to be approximated

    1−cos⁡(x)xsin⁡(x)\frac{1 - \cos{\left(x \right)}}{x \sin{\left(x \right)}}

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

  2. Substitute the small-angle approximations

    x22x2\frac{\frac{x^{2}}{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. State the approximation

    1−cos⁡(x)xsin⁡(x)≈12\frac{1 - \cos{\left(x \right)}}{x \sin{\left(x \right)}}\approx \frac{1}{2}

    This is the required small-angle approximation.

Answer
12\frac{1}{2}
Question 5
2 markseasy
Use small-angle approximations to simplify cos⁡(2x)\cos{\left(2 x \right)} for small xx.

Worked solution

  1. Write the expression to be approximated

    cos⁡(2x)\cos{\left(2 x \right)}

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

  2. Substitute the small-angle approximations

    1−2x21 - 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. State the approximation

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

    This is the required small-angle approximation.

Answer
1−2x21 - 2 x^{2}

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