Double Angle Identities on the Unit Circle
See how sin(2θ) and cos(2θ) follow from coordinates on the unit circle and standard identity proofs.
Quick Answer
- Short definition
- On the unit circle, doubling θ moves the point from (cos θ, sin θ) to (cos 2θ, sin 2θ), which identities express algebraically.
- Formula
- cos(2θ) = cos² θ − sin² θ
Introduction
Double Angle Calculator complements unit-circle study by giving exact numeric outputs for any θ you enter. The circle picture and the identities describe the same facts in two languages: geometry and algebra.
This article focuses on coordinates. You will see how sin(2θ) and cos(2θ) follow from the point at θ and the point at 2θ on the circle of radius 1.
Pair this page with Double Angle in Trigonometry for triangle and graph connections, and with Double Angle Identities for proof-style simplification beyond coordinates.
Main Content
What is it?
The unit circle radius is 1, so cos θ and sin θ are the x and y coordinates at angle θ. At angle 2θ, the coordinates are cos 2θ and sin 2θ. Doubling the angle moves the point along the circle, sometimes into a new quadrant.
Double-angle identities encode that move without redrawing the circle each time. They support proofs and quick mental checks in trigonometry courses. When you forget a sign, returning to the circle often resolves the mistake faster than re-deriving algebra.
The sine identity sin(2θ) = 2 sin θ cos θ has a coordinate interpretation: the y-value at 2θ equals twice the product of the x- and y-values at θ. The cosine identity cos(2θ) = cos² θ − sin² θ compares the x-value at 2θ to squared coordinates at θ.
Reference angles still apply at 2θ. If 2θ lands in Quadrant II, cosine is negative and sine is positive, regardless of where θ started. Identities preserve those signs automatically when you substitute exact values.
Formula
sin(2θ) = 2 sin θ cos θ links the y-coordinate at 2θ to products of coordinates at θ.
cos(2θ) = cos² θ − sin² θ links the x-coordinate at 2θ to squared coordinates at θ.
Equivalent forms cos(2θ) = 2 cos² θ − 1 and cos(2θ) = 1 − 2 sin² θ follow from cos² θ + sin² θ = 1. Use whichever matches the coordinates you know at θ.
For a broader list and selection tips, see Double Angle Formula. For algebraic uses outside the circle diagram, see Double Angle Identities.
Step-by-step guide
- Plot θ on the unit circle. Mark (cos θ, sin θ) and note the quadrant.
- Plot 2θ on the same circle. Mark (cos 2θ, sin 2θ) and note whether the quadrant changed.
- Apply identities to connect the pairs. Write the identity before substituting coordinates.
- Verify with the calculator. Enter θ in the same unit as your sketch.
- Connect to graph ideas. Read Double Angle in Trigonometry for how the same 2θ pattern changes period on sine and cosine graphs.
Example
Example 1: θ = 45° gives (cos θ, sin θ) = (√2/2, √2/2).
2θ = 90° gives (0, 1). cos(90°) = 0 matches cos² 45° − sin² 45° = 1/2 − 1/2 = 0.
Example 2: θ = 60° → (1/2, √3/2). 2θ = 120° → (−1/2, √3/2).
cos(120°) = −1/2 = 2(1/2)² − 1. The negative x-coordinate at 2θ appears correctly in the identity result.
Example 3: θ = π/6 rad. sin(π/3) = √3/2 = 2(1/2)(√3/2). The y-coordinate doubled through the product rule matches the point at 2θ.
FAQ
Why learn the unit circle view if I have formulas?
The circle view explains signs and quadrant behavior, which formulas alone do not show as clearly when you work quickly on a test.
Does the radius matter for double-angle identities?
The standard identities assume the unit circle. For a circle of radius r, coordinates scale by r, but the angle relationships and identity forms stay the same.
Conclusion
Pair visual unit-circle reasoning with identity algebra for a complete understanding of double angles. Sketch when signs confuse you; use formulas when speed matters.
Practice coordinates with Double Angle Examples, then explore applied models in Double Angle Applications.