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julsineya [31]
2 years ago
6

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

Mathematics
1 answer:
kotegsom [21]2 years ago
8 0

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}

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