Lesson Worksheet: Small-Angle Approximations Mathematics

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In this worksheet, we will practice approximating trigonometric functions when the angle of the function is close to zero.

Q1:

What is the approximate value of sintantan2𝑥−𝑥+𝑥2𝑥 for small values of 𝑥 measured in radians?

Q2:

What is the approximate value of cossin2𝑥−1𝑥 for small values of 𝑥 measured in radians?

  • A−4
  • B−2+2𝑥
  • C−2
  • D2−2𝑥
  • E2𝑥−1𝑥

Q3:

By first showing that when 𝜃 is a small angle measured in radianssincostan2𝜃−4𝜃+𝜃≈8𝜃+3𝜃−1, state the approximate value of sincostan2𝜃−4𝜃+𝜃 for small values of 𝜃 measured in radians.

Q4:

What is the approximate value of sintantan𝑥−5𝑥2𝑥 for small values of 𝑥 measured in radians?

Q5:

Find an expression for the approximate value of sintancos2𝑥+𝑥+12𝑥+2𝑥, given that 𝑥 is a small angle measured in radians.

  • A3−4𝑥
  • B3𝑥+1
  • C3𝑥+14𝑥+1
  • D3𝑥
  • E2𝑥+1

Q6:

Given that 𝜃 is a small angle measured in radians, find the values of 𝑎 and 𝑏 such that 22𝜃+2𝜃𝜃−1≈𝑎𝜃+𝑏cossintan.

  • A𝑎=−1, 𝑏=1
  • B𝑎=−2, 𝑏=1
  • C𝑎=−4, 𝑏=2
  • D𝑎=−2, 𝑏=−2
  • E𝑎=−4, 𝑏=−2

Use your approximation to give an approximate numerical value of 220.01+20.010.01−1cossintan.

Q7:

By first calculating cos(0.35) and then approximating cos(0.35) using the approximation for cos(𝜃) for an angle measured in radians, calculate the percentage error in the approximation for cos(0.35) to two significant figures.

Q8:

The percentage error in the approximation for sin(0.25) is approximately 1.05%.

By using the exact value of sin(0.2), find the percentage error in approximating this value using the small angle approximation. Give the answer to two significant figures. Assume all angles are in radians.

Which of the following is the reason for the difference between the two percentage errors?

  • AThe larger the angle, the smaller the error will be.
  • BThe smaller the angle, the smaller the error will be.
  • CThere is no relationship between the errors.

Q9:

Given that the percentage error for an approximation of tan𝑥 for a small positive angle 𝑥 measured in radians is 2%, which of the following is true?

  • A48𝑥≈51𝑥tan
  • B50𝑥≈51𝑥tan
  • C51𝑥≈49𝑥tan
  • D50𝑥≈49𝑥tan
  • E49𝑥≈50𝑥tan

Q10:

Find an expression for the approximate value of −52𝑥+12𝑥+7𝑥+3costansin, given that 𝑥 is a small angle measured in radians.

  • A5𝑥−19𝑥+2
  • B5𝑥+19𝑥+2
  • C24𝑥−19𝑥−2
  • D10𝑥+19𝑥−2
  • E10𝑥+19𝑥+3

By setting this expression equal to 0 and solving, find the approximate solutions to −52𝑥+12𝑥+7𝑥+3costansin and comment on the validity of the solutions.

  • A𝑥=0.1 and 𝑥=−3 are approximate valid solutions.
  • B𝑥=0.1 and 𝑥=−2 are approximate valid solutions.
  • C𝑥=−0.1 is the only approximate valid solution since the small angle approximations are only valid for small angles.
  • D𝑥=0.1 is the only approximate valid solution since the small angle approximations are only valid for small angles.
  • EThere are no solutions.

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