Triple Angle Formulas
sin3θ and cos3θ in terms of θ.
Key idea
sin3θ and cos3θ in terms of θ.
In depth
sin 3θ = 3 sin θ − 4 sin³θ and cos 3θ = 4cos³θ − 3 cos θ. Derive them by writing 3θ = 2θ + θ and using the sum and double angle formulas.
Worked example
If sin θ = 1/2, sin 3θ = 3/2 − 4/8 = 1. Indeed 3 × 30° = 90°.
Formula
Real-world uses
- Solving cubic trig equations in exams and studying harmonics in music and electronics
Common mistake
Confusing the two with each other’s signs.
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 Triple Angle Formulas?
sin3θ and cos3θ in terms of θ. It is a topic at the expert level in Trigonometry.
What is the formula for Triple Angle Formulas?
The key formula is sin3θ = 3sinθ − 4sin³θ.
Where is Triple Angle Formulas used in real life?
Real-world examples: Solving cubic trig equations in exams and studying harmonics in music and electronics.
What should I learn before Triple Angle Formulas?
Make sure you are comfortable with Sum-to-Product Formulas, the previous topic in Trigonometry.
What is a common mistake with Triple Angle Formulas?
Confusing the two with each other’s signs.
How can I practise Triple Angle Formulas?
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 Triple Angle Formulas?
Move on to Proving Trig Identities 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.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).
- Trig Equations with Identities – Use identities to factor trig equations.