Question

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

Answer 1

Step-by-step explanation:

Ruler's formula states that

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

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

$2\cos{(\theta_1+\theta_2)} = e^{i(\theta_1+\theta_2)} + e^{-i(\theta_1+\theta_2)}$

or

$\cos{(\theta_1+\theta_2)} = \dfrac{e^{i(\theta_1+\theta_2)} + e^{-i(\theta_1+\theta_2)}}{2}$

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

$2i\sin{(\theta_1+\theta_2)} = e^{i(\theta_1+\theta_2)} - e^{-i(\theta_1+\theta_2)}$

or

$\sin{(\theta_1+\theta_2)} = \dfrac{e^{i(\theta_1+\theta_2)} - e^{-i(\theta_1+\theta_2)}}{2i}$