Question

Using Euler’s formular obtain trigonometric formulars for cos(1 + 2) and sin(1 + 2).

Answer 1

Explanation:

Ruler's formula states that

`e^(i\theta) = cos(\theta) +isin(\theta)`

We also know that

`e^(i\theta_1) \cdot e^(i\theta_2) = e^(i(\theta_1+\theta_2))`

therefore,

`e^(i(\theta_1+\theta_2))= cos((\theta_1+\theta_2))+ sin((\theta_1+\theta_2))` (1)

Similarly, we can write

`e^(-i(\theta_1+\theta_2)) = cos((\theta_1+\theta_2)) - sin((\theta_1+\theta_2))` (2)

Adding Eqn(1) and Eqn(2) together, we get

`2cos((\theta_1+\theta_2)) = e^(i(\theta_1+\theta_2)) + e^(-i(\theta_1+\theta_2))`

or

`cos((\theta_1+\theta_2)) = (e^(i(\theta_1+\theta_2)) + e^(-i(\theta_1+\theta_2)))/(2)`

To get the expression for the sine function, we subtract Eqn(2) from Eqn(1) to get

`2isin((\theta_1+\theta_2)) = e^(i(\theta_1+\theta_2)) - e^(-i(\theta_1+\theta_2))`

or

`sin((\theta_1+\theta_2)) = (e^(i(\theta_1+\theta_2)) - e^(-i(\theta_1+\theta_2)))/(2i)`