A Geometric Way to Visualise sin(x + y) and cos(x + y)

The angle–addition identities for sine and cosine often appear as algebraic formulas, but they can also be understood by combining two right triangles in a simple geometric construction. The calculations for the side lengths follow directly from the definitions of sine and cosine.

sin(x + y) = sin x cos y + cos x sin y
cos(x + y) = cos x cos y − sin x sin y

The angle–addition identities for sine and cosine.

Start with a right triangle of angle y and hypotenuse 1. From basic trigonometry, its horizontal and vertical sides are:

cos y and sin y.

Next, attach a second right triangle with angle x. Its hypotenuse is the side of length cos y from the first triangle, so its adjacent and opposite sides become:

  • adjacent = cos x · cos y
  • opposite = sin x · cos y

Likewise, if the first triangle's vertical side sin y is used as a hypotenuse in a similar way, it contributes:

  • adjacent = cos x · sin y
  • opposite = sin x · sin y

When the horizontal components are combined, they give the expression for cos(x + y). When the vertical components are combined, they produce sin(x + y). The familiar angle–addition identities emerge naturally from these two constructions.

This idea comes from a classic diagram in Proofs Without Words 2 by Roger B. Nelsen, with additional workings and annotations added for clarity.

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