Sine and Cosine Series
Infinite series for sin and cos.
Key idea
Infinite series for sin and cos.
In depth
sin x = x − x³/3! + x⁵/5! − ... and cos x = 1 − x²/2! + x⁴/4! − ... They converge for all x, and let you compute trig values with just arithmetic.
Worked example
sin(0.5) ≈ 0.5 − 0.0208 + 0.00026 ≈ 0.4794.
Formula
Real-world uses
- Approximating sine and cosine in computer code and in physics derivations
Common mistake
Using degrees instead of radians in the series.
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 Sine and Cosine Series?
Infinite series for sin and cos. It is a topic at the expert level in Trigonometry.
What is the formula for Sine and Cosine Series?
The key formula is sin x = x − x³/3! + x⁵/5! − ….
Where is Sine and Cosine Series used in real life?
Real-world examples: Approximating sine and cosine in computer code and in physics derivations.
What should I learn before Sine and Cosine Series?
Make sure you are comfortable with Trig Values via Pythagoras Triples, the previous topic in Trigonometry.
What is a common mistake with Sine and Cosine Series?
Using degrees instead of radians in the series.
How can I practise Sine and Cosine Series?
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 Sine and Cosine Series?
Move on to Law of Tangents 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).