Small Angle Approximation
For small θ, sinθ ≈ θ and tanθ ≈ θ.
Key idea
For small θ, sinθ ≈ θ and tanθ ≈ θ.
In depth
For small θ in radians, sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − θ²/2. They come from the first terms of the Taylor series. Used in pendulums and optics.
Worked example
sin(0.1) = 0.0998 ≈ 0.1. Error about 0.0002.
Formula
Real-world uses
- Estimating pendulum periods and small deflections in engineering and optics
Common mistake
Using it with degrees. The angle must be in radians.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Small Angle Approximation?
For small θ, sinθ ≈ θ and tanθ ≈ θ. It is a topic at the expert level in Trigonometry.
What is the formula for Small Angle Approximation?
The key formula is sinθ ≈ θ.
Where is Small Angle Approximation used in real life?
Real-world examples: Estimating pendulum periods and small deflections in engineering and optics.
What should I learn before Small Angle Approximation?
Make sure you are comfortable with Trig Substitution Preview, the previous topic in Trigonometry.
What is a common mistake with Small Angle Approximation?
Using it with degrees. The angle must be in radians.
How can I practise Small Angle Approximation?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Small Angle Approximation?
Move on to Chebyshev Polynomials next.
Related expert topics in Trigonometry
- Half Angle Formulas – Trig of θ/2.
- Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas – Convert products into sums.
- Sum-to-Product Formulas – Convert sums into products.
- Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).