Geometric definitions and trigonometric identities

Idea

The angle $\theta$ picks the point $P$ on the unit circle. The horizontal coordinate is $\cos \theta$, the vertical coordinate is $\sin \theta$, and $\tan \theta =\sin \theta /\cos \theta$ when $\cos \theta \ne 0$. The identity ${\sin }^2\theta +{\cos }^2\theta =1$ is $P$ staying on the circle of radius $1$ centered at the origin.

The same functions then play three roles: coordinates on the unit circle, side-and-angle relations in a triangle, and the algebra of adding two angles.

Interaction

• Circle to definition: drag Angle θ and watch $P(\theta )$ move with $\sin \theta$, $\cos \theta$, and $\tan \theta$. When $\cos \theta =0$, tangent is undefined
• Angle to triangle: drag vertices A, B, and C. The side lengths, the angles, and the circumradius line up with the sine law and the cosine law. Reset triangle restores the starting triangle
• Rotation to formula: drag Angle α and Angle β and watch the sum angle $\alpha +\beta$ against the angle-addition formulas for $\sin (\alpha +\beta )$ and $\cos (\alpha +\beta )$
• Advanced guide: on: show the extra guides. In the rotation view the formula list adds $2\sin \alpha \cos \beta =\sin (\alpha +\beta )+\sin (\alpha -\beta )$

Suggested order: Circle to definition, then Angle to triangle, then Rotation to formula.

What to notice

• The same angle is a pair of coordinates on the unit circle, and then a ratio of sides in a triangle. Both readings use $\sin$ and $\cos$.
• Solving the triangle carries the angle from the circle onto the side lengths. The circumcircle carries that ratio back onto a circle.
• Adding two angles is still walking an arc on the circle, first $\alpha$ and then $\beta$. That sum arc matches the angle-addition formulas. The listed value $\cos 2\alpha$ is twice $\alpha$, not that sum, unless $\beta =\alpha$.

Related

• Unit circle and trigonometric definitions
• Sine law and cosine law
• Trigonometric identities and angle sums

Further reading

• Trigonometric functions (Wikipedia)
• List of trigonometric identities (Wikipedia)
• Law of sines (Wikipedia)
• Law of cosines (Wikipedia)