Double-Angle Identities
Formulas for sin 2θ and cos 2θ.
About this topic
Setting A = B in the sum formulas gives the double-angle identities: sin 2θ = 2 sin θ cos θ, and cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. Rearranged, they become the half-angle identities.
The three forms of cosine are interchangeable, and choosing the best one depends on what you know. If only sine is given, use 1 − 2sin²θ; if only cosine is given, use 2cos²θ − 1. Tangent has tan 2θ = 2 tan θ/(1 − tan²θ). Solving the cosine forms for the squared terms gives sin²θ = (1 − cos 2θ)/2 and cos²θ = (1 + cos 2θ)/2.
Example: if sin θ = 3/5 and θ is acute, then cos θ = 4/5, so sin 2θ = 2(3/5)(4/5) = 24/25 and cos 2θ = 16/25 − 9/25 = 7/25.
They simplify expressions, solve trigonometric equations and are used in calculus to integrate squared trig functions. In physics, the range of a projectile launched at angle θ is proportional to sin 2θ, so it is largest at 45°. A common mistake is thinking sin 2θ = 2 sin θ; the factor 2 does not simply pull out.
Key ideas
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos²θ − sin²θ
- Also 1 − 2sin²θ
Quick check
Use cases: where Double-Angle Identities helps
- Homework: review the key idea behind Double-Angle Identities before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Double-Angle Identities without notes.
- Visual learning: play with the interactive lab to see how Double-Angle Identities changes when you adjust its inputs.
- Teaching a friend: explain Double-Angle Identities in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Double-Angle Identities?
Formulas for sin 2θ and cos 2θ.
What will I learn in Double-Angle Identities?
sin 2θ = 2 sin θ cos θ. cos 2θ = cos²θ − sin²θ. Also 1 − 2sin²θ.
Is Double-Angle Identities part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Double-Angle Identities lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Double-Angle Identities?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.