Trigonometric Addition Formulas

1Introduction

The addition formulas express in coordinates the fact that composing two plane rotations gives one rotation through the sum of their angles. This lecture derives them rigorously from rotation matrices and organizes the difference-angle, double-angle, and half-angle formulas as consequences.

2Rotation Matrices

For a $2\times 2$2\times2 matrix $A=\left(a_{ij}\right)$A=(a_{ij}) and a column vector $(x,y{)}^{\mathsf{T}}$(x,y)^\mathsf T, define

For two matrices $A,B$A,B, define $(AB{)}_{ij}=\sum _{k=1}^2{a}_{ik}{b}_{kj}$(AB)_{ij}=\sum_{k=1}^2a_{ik}b_{kj}. This product represents composition: first apply $B$B, then apply $A$A. Write a point at distance $r\ge 0$r\geq0 and with argument $t$t as $r{P}_t$rP_t, and define rotation by $Q_{\theta }\left(r{P}_t\right)=r{P}_{t+\theta }$Q_\theta(rP_t)=rP_{t+\theta}. Addition of signed arc lengths gives $Q_{\alpha }\circ Q_{\beta }=Q_{\alpha +\beta }$Q_\alpha\circ Q_\beta=Q_{\alpha+\beta}. A rotation fixes the origin and preserves both parallelograms spanned by vectors and directed distances along lines; hence it preserves vector addition and real scalar multiplication. Thus $Q_{\theta }$Q_\theta is linear.

Let $e_1=(1,0{)}^{\mathsf{T}}$e_1=(1,0)^\mathsf T and $e_2=(0,1{)}^{\mathsf{T}}$e_2=(0,1)^\mathsf T be the standard basis. The map $Q_{\theta }$Q_\theta is linear and preserves lengths, orthogonality, and orientation. It sends $e_1$e_1 to $u=(\cos \theta ,\sin \theta {)}^{\mathsf{T}}$u=(\cos\theta,\sin\theta)^\mathsf T. The image of $e_2$e_2 must be the unique unit vector perpendicular to $u$u for which $(u,v)$(u,v) has positive orientation, namely $v=(-\sin \theta ,\cos \theta {)}^{\mathsf{T}}$v=(-\sin\theta,\cos\theta)^\mathsf T. Hence the matrix of $Q_{\theta }$Q_\theta is

Applying $R_{\beta }$R_\beta and then $R_{\alpha }$R_\alpha geometrically rotates every vector through a total angle $\alpha +\beta$\alpha+\beta. Thus, without using the addition formulas,

Direct multiplication gives

Comparison with $R_{\alpha +\beta }$R_{\alpha+\beta} proves, for all real $\alpha ,\beta$\alpha,\beta,

3Difference-Angle Formulas

Replace $\beta$\beta by $-\beta$-\beta and use $\cos (-\beta )=\cos \beta$\cos(-\beta)=\cos\beta and $\sin (-\beta )=-\sin \beta$\sin(-\beta)=-\sin\beta:

4Double-Angle Formulas

Setting $\beta =\alpha =\theta$\beta=\alpha=\theta gives

The last two forms use ${\sin }^2\theta +{\cos }^2\theta =1$\sin^2\theta+\cos^2\theta=1.

5Addition Formulas for Tangent

If $\cos \alpha \cos \beta \ne 0$\cos\alpha\cos\beta\ne0 and $1-\tan \alpha \tan \beta \ne 0$1-\tan\alpha\tan\beta\ne0, divide the sine addition formula by the cosine addition formula to obtain

Similarly, if $\cos \alpha \cos \beta \ne 0$\cos\alpha\cos\beta\ne0 and $1+\tan \alpha \tan \beta \ne 0$1+\tan\alpha\tan\beta\ne0, then

6Half-Angle Formulas and Signs

Replacing $\theta$\theta by $\theta /2$\theta/2 in the double-angle formulas gives

Taking square roots yields

where the sign is determined by the position of $\theta /2$\theta/2, including both quadrants and coordinate axes. Formulas involving tangent apply only where their denominators are nonzero.

7An Alternative Proof Using Complex Numbers

This section is an alternative proof to be used only after complex numbers and Euler's formula have been introduced; it is not a prerequisite for the rotation-matrix proof.

Substitute $e^{i\theta }=\cos \theta +i\sin \theta$e^{i\theta}=\cos\theta+i\sin\theta into

and compare real and imaginary parts to obtain the same addition formulas.

8Scope

The sine and cosine formulas for sums, differences, and double angles hold for all real angles. The tangent formulas hold under the nonzero conditions stated above; every transformation involving a quotient requires a nonzero denominator. When square roots are taken in the half-angle formulas, the signs must be selected from the position of the half-angle, including both quadrants and coordinate axes.

9Related Lectures