Two unit-amplitude waves shifted by π/3 combine with amplitude √3, not 2. Angle-addition identities describe rotations and wave reinforcement. Use products, the circle and sinusoids.

The unit circle satisfies x²+y²=1; substitute cos/sin coordinates. Reflection gives negative-angle signs:

(sin⁡)^(2)x + (cos⁡)^(2)x = 1; sin⁡( − x) = − sin⁡x, cos⁡( − x) = cos⁡x.

Likewise sin(π−x)=sin x and cos(π−x)=−cos x: the horizontal coordinate reverses in quadrant II.

Geometric derivation

Rotate two perpendicular unit directions by α: (cosα,sinα) and (−sinα,cosα). A unit vector (an arrow of length 1) at additional angle β has projections cosβ and sinβ on them. Add horizontal and vertical contributions:

cos⁡(α + β) = cos⁡αcos⁡β − sin⁡αsin⁡β.

sin⁡(α + β) = sin⁡αcos⁡β + cos⁡αsin⁡β.

Replace β by −β for difference identities, or by α for double angles:

sin⁡2x = 2sin⁡xcos⁡x; cos⁡2x = (cos⁡)^(2)x − (sin⁡)^(2)x.

Two successive rotations

x²+y²=1

-101-11αα+βxy

Sum of shifted waves

sin t + sin(t+π/3) = √3 sin(t+π/6)

0π/2π3π/22π-202t, rady
Rotations α=π/6 and β=π/4 give 5π/12. Below, the solid sum of two dashed waves has amplitude √3. Uniform scales; equal circle scales; radians.

Combining waves

Use α=t+π/6, β=π/6 in sum and difference identities:

sin⁡t + sin⁡(t + π/3) = √(3)sin⁡(t + π/6).

Generally sin t+sin(t+δ)=2cos(δ/2)sin(t+δ/2); amplitude is the coefficient’s absolute value. δ=π gives cancellation in this ideal model. OpenStax

Branch · IdentitiesOptional · may be skipped

Triple and half angles, sums and products

Repeated addition or half-angle substitution gives these optional identities:

sin⁡3x = 3sin⁡x − 4(sin⁡)^(3)x; cos⁡3x = 4(cos⁡)^(3)x − 3cos⁡x.

(sin⁡)^(2)(x/2) = (1 − cos⁡x)/(2); (cos⁡)^(2)(x/2) = (1 + cos⁡x)/(2).

When taking a square root, the quadrant determines the sign of sin(x/2) or cos(x/2); do not always choose plus.

sin⁡u + sin⁡v = 2sin⁡((u + v)/(2))cos⁡((u − v)/(2)); cos⁡u + cos⁡v = 2cos⁡((u + v)/(2))cos⁡((u − v)/(2)).

2sin⁡ucos⁡v = sin⁡(u + v) + sin⁡(u − v);

2cos⁡ucos⁡v = cos⁡(u + v) + cos⁡(u − v); 2sin⁡usin⁡v = cos⁡(u − v) − cos⁡(u + v).

Divide the sine addition formula by the cosine formula:

tan⁡(α + β) = (tan⁡α + tan⁡β)/(1 − tan⁡αtan⁡β).

This requires cosα, cosβ and cos(α+β) all nonzero. Sine and cosine identities have no such exceptions. Preserve the zero case before canceling a trigonometric factor or solutions can be lost.