using eulers formula what are the trigonometric identities for (cosθ1 + cosθ2) and (sinθ1 + sinθ2)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember the transformation of the sum of trigonometric functions in products.

`cos theta_1 + cos theta_2 = 2 cos ((theta_1 + theta_2)/2)*cos ((theta_1- theta_2)/2)`

Use Euler's formula to express the cosine function.

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

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

Use the exponential properties:

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

Transform the sum `sin theta_1 + sin theta_2 = 2sin ((theta_1 + theta_2)/2)*cos ((theta_1- theta_2)/2)`

Use Euler's formula to express the sine function.

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

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

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

ANSWER: These are the results obtained using Euler's transformations:

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

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

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