Trigonometric identities and angle sums

Parametric equations

The angles $\alpha$ and $\beta$ are two directions on the unit circle. A sum-to-product identity rewrites a sum or a difference as a product in the midpoint angle $m=\frac{\alpha +\beta }{2}$ and the offset $d=\frac{\alpha -\beta }{2}$. For example,

The angle-addition formula $\sin (\alpha +\beta )$ and the double-angle formula $\sin 2\alpha$ are other algebraic readings of the same angle composition. This figure stays with the four sum-to-product identities, and with the product-to-sum reading in the other direction.

Interaction

• sin+: $\sin \alpha +\sin \beta =2\sin m\cos d$. The sum shows the mean direction
• sin−: $\sin \alpha -\sin \beta =2\cos m\sin d$. The difference shows the half-difference vector
• cos+: $\cos \alpha +\cos \beta =2\cos m\cos d$. The sum shows the mean direction
• cos−: $\cos \alpha -\cos \beta =-2\sin m\sin d$. The difference shows the half-difference vector
• Angle α: drag this endpoint on the circle, or use the slider
• Angle β: drag the other endpoint, or use the slider
• Composition guide: on: mark $m=\frac{\alpha +\beta }{2}$ and $d=\frac{\alpha -\beta }{2}$
• Reading: sum-to-product: read the same figure forward as sum-to-product, or backward as product-to-sum
• Angle display: degrees: show the angles in degrees or in radians

What to notice

• Sum-to-product comes from the midpoint angle and the offset of the two directions. It is not a list of unrelated formulas.
• When $\alpha \approx \beta$, the offset $d$ goes to $0$, and the figure marks $\alpha \approx \beta$.
• Product-to-sum and sum-to-product are the forward and backward readings of the same geometric relation.

Related

• Unit circle and trigonometric definitions
• Sine law and cosine law
• Geometric definitions and trigonometric identities

Further reading

• List of trigonometric identities (Wikipedia)