Small-angle reasoning
Objective. Use and critique small-angle approximations in radians.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Small-angle approximations use sin θ≈θ, tan θ≈θ, and cos θ≈1−θ²/2 for angles measured in radians close to zero, derived from the early terms of each function's series behaviour near the origin. These approximations let physics and engineering problems avoid full trigonometric evaluation when angles are small, such as pendulum motion. For θ=0.1 radians, sin θ≈0.0998 and the approximation 0.1 is accurate to three decimal places, showing the method's validity near zero. Always state that θ must be in radians and reasonably small, since the approximation breaks down for larger angles.
Worked example
Use the small-angle approximation to estimate the value of tan θ when θ = 0.05 radians.
- 1Recall the small-angle approximation tan θ≈θ for small θ in radians
- 2Confirm θ=0.05 radians is sufficiently small for the approximation
- 3Substitute directly into the approximation
- 4State the estimate as tan θ≈0.05
Answer: ≈0.05
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Practice
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Start practisingCommon mistake. Students apply small-angle approximations to angles given in degrees without converting to radians first; sin 10°≈10 is wrong, since the approximation only works once 10° is converted to approximately 0.1745 radians.
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