5.2.6

Small Angle Approximation

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Small Angle Approximation

We can approximate the values of trigonometric functions when the angle is very small in order to simplify calculations. We call these the small angle approximations.

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Sine approximation

  • When the angle θ\theta is close to zero and measured in radians, we can approximate the sine of this angle as:
    • sinθθ\sin\theta \approx \theta
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Cosine approximation

  • When the angle θ\theta is close to zero and measured in radians, we can approximate the cosine of this angle as:
    • cosθ1θ22\cos\theta \approx 1-\frac{\theta^2}{2}
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Tangent approximation

  • When the angle θ\theta is close to zero and measured in radians, we can approximate the tangent of this angle as:
    • tanθθ\tan\theta \approx \theta
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Example

  • What is the approximation of the following expression when θ\theta is small?
    • cos4θ+sin3θ\cos4\theta + \sin3\theta
  • What is the approximate value of the expression when θ\theta is small?
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Approximations

  • When θ\theta is small, cosθ1θ22\cos\theta \approx 1-\frac{\theta^2}{2} and sinθθ\sin\theta \approx \theta.
  • That means we can say that:
    • cos4θ1(4θ)22\cos4\theta\approx 1 - \frac{(4\theta)^2}{2}
    • sin3θ3θ\sin3\theta \approx 3\theta
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Substitute approximations in

  • Substituting these approximations in, we have:
    • cos4θ+sin3θ1(4θ)22+3θ\cos4\theta + \sin3\theta\approx 1-\frac{(4\theta)^2}{2}+3\theta
  • Simplifying this expression gives:
    • cos4θ+sin3θ18θ2+3θ\cos4\theta + \sin3\theta\approx 1 - 8\theta^2 + 3\theta
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Approximate value

  • When θ\theta is small, any terms with factors of θ\theta will also be small.
  • That means we can disregard these terms when finding the value of the expression:
    • cos4θ+sin3θ1\cos4\theta + \sin3\theta\approx 1

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2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

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