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

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Step-by-step explanation:

Ruler's formula states that

[tex]e^{i\theta} = \cos{\theta} +i\sin{\theta}[/tex]

We also know that

[tex]e^{i\theta_1} \cdot e^{i\theta_2} = e^{i(\theta_1+\theta_2)}[/tex]

therefore,

[tex]e^{i(\theta_1+\theta_2)}= \cos{(\theta_1+\theta_2)}+ \sin{(\theta_1+\theta_2)}[/tex] (1)

Similarly, we can write

[tex]e^{-i(\theta_1+\theta_2)} = \cos{(\theta_1+\theta_2)} - \sin{(\theta_1+\theta_2)}[/tex] (2)

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

[tex]2\cos{(\theta_1+\theta_2)} = e^{i(\theta_1+\theta_2)} + e^{-i(\theta_1+\theta_2)}[/tex]

or

[tex]\cos{(\theta_1+\theta_2)} = \dfrac{e^{i(\theta_1+\theta_2)} + e^{-i(\theta_1+\theta_2)}}{2}[/tex]

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

[tex]2i\sin{(\theta_1+\theta_2)} = e^{i(\theta_1+\theta_2)} - e^{-i(\theta_1+\theta_2)}[/tex]

or

[tex]\sin{(\theta_1+\theta_2)} = \dfrac{e^{i(\theta_1+\theta_2)} - e^{-i(\theta_1+\theta_2)}}{2i}[/tex]

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