Home | 18.013A | Chapter 2 | Section 2.3 | | Tools Glossary Index Up Previous Next

Exercise 2.10

Derive the formulae for sin ⁡ x and cos ⁡ x in terms of exp ⁡ i x and exp ⁡ ( − i x ) , that follows from equation (A) above.

Solution:

Equation (A) reads

exp ⁡ i x = cos ⁡ x + i sin ⁡ x

and it holds for all x positive or negative, so that we also have

exp ⁡ ( − i x ) = cos ⁡ x − i sin ⁡ x

where we have used the fact that cosine is an even function and sine an odd one.

We can solve these two equations for cos ⁡ x and sin ⁡ x

cos ⁡ x = exp ⁡ i x + exp ⁡ ( − i x ) 2 sin ⁡ x = exp ⁡ i x − exp ⁡ ( − i x ) 2 i

Up Previous Next