Even-Odd Identities
Cosine is even; sine and tangent are odd.
Key idea
Cosine is even; sine and tangent are odd.
In depth
Cosine and secant are even: cos(−θ) = cos θ. Sine, tangent, cosecant, cotangent are odd: sin(−θ) = −sin θ. They come from reflecting a point across the x-axis on the unit circle.
Worked example
sin(−30°) = −1/2 and cos(−30°) = √3/2.
Formula
Real-world uses
- Simplifying expressions such as cos(−θ) in physics and signal processing
Common mistake
Treating sine as even.
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 Even-Odd Identities?
Cosine is even; sine and tangent are odd. It is a topic at the advanced level in Trigonometry.
What is the formula for Even-Odd Identities?
The key formula is sin(−θ) = −sinθ.
Where is Even-Odd Identities used in real life?
Real-world examples: Simplifying expressions such as cos(−θ) in physics and signal processing.
What should I learn before Even-Odd Identities?
Make sure you are comfortable with Cofunction Identities, the previous topic in Trigonometry.
What is a common mistake with Even-Odd Identities?
Treating sine as even.
How can I practise Even-Odd Identities?
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 Even-Odd Identities?
Move on to Sum and Difference Formulas next.
Related advanced topics in Trigonometry
- Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
- Phase Shift – Horizontal translation of a wave.
- Vertical Shift and Midline – Moves the wave up or down.
- General Sinusoid – Combine amplitude, period, phase and shift.
- Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
- Pythagorean Identity – The key identity from the unit circle.