Saturday, January 20, 2024

Trigonometry+ 01: UNIZOR.COM - Math+ & Problems - Trigonometry

Notes to a video lecture on

Trigonometry+ 01

Problem A

Using the Euler Formula
ei·x = cos(x)+i·sin(x)
prove the formulas for sine and cosine of a sum of two angles.

Solution A

The Euler formula states that the real part of ei·x is cos(x) and its imaginary part s i·sin(x)

From i² = −1,
ei·(α+β) = cos(α+β)+i·sin(α+β),
and a(p+q) = ap·aq
cos(α+β)+i·sin(α+β) =
= ei·(α+β) = ei·α·ei·β =
[cos(α)+i·sin(α)] ·
[cos(β)+i·sin(β)] =
= cos(α)·cos(β)−sin(α)·sin(β) +
+ i·cos(α)·sin(β)+i·sin(α)·cos(β)

The real part of the last expression is
cos(α)·cos(β) − sin(α)·sin(β)
Its imaginary part is
[cos(α)·sin(β) + sin(α)·cos(β)]

(the real part of ei·(α+β))
equals to
cos(α)·cos(β) − sin(α)·sin(β),
cos(α+β) =
= cos(α)·cos(β) − sin(α)·sin(β)

(the imaginary part of ei·(α+β))
equals to
[cos(α)·sin(β) + sin(α)·cos(β)],
sin(α+β) =
= cos(α)·sin(β) + sin(α)·cos(β)

Problem B

Using the Euler Formula
ei·x = cos(x)+i·sin(x)
prove the formulas for derivatives of sine and cosine functions.

Solution B

Let's differentiate the Euler's formula
ei·x = cos(x)+i·sin(x)

On the left side the result is
d/dx[ei·x] = i·ei·x =
= i·
[cos(x) + i·sin(x)] =
= −sin(x) + i·cos(x)

On the right side the result of differentiation is
d/dx[cos(x)] + i·d/dx[sin(x)]

The results of differentiation of left and right sides of the Euler's formula must be equal to each other:
d/dx[cos(x)] + i·d/dx[sin(x)] =
= −sin(x) + i·cos(x)

When two complex numbers are equal to each other, their real and imaginary parts must be equal.
d/dx[cos(x)] = −sin(x)
d/dx[sin(x)] = cos(x)

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