1.9 References of Trigonometric Identities |
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(Form 4) pþðx, tÞ ¼ 21 ð 2 þ j3Þe jð4t 5xÞ |
þ ð 2 j3Þe jð4t 5xÞ |
Exercise 1.7
Express the following simple harmonic motion in the four equivalent forms:
p 3π xðtÞ ¼ cos 5t 4 2 sin 5t þ 4
(Answers): (You must show all your work for full credit!)
(Form 1) ¼ 3 cos 5t + 4 sin 5t (Form 2) ¼5 cos (5t 2.214)
(Form 3) ¼ 12 5e jð5t 2:214Þ þ 5e jð5t 2:214Þ
(Form 4) ¼ 12 ð 3 j4Þe j5t þ ð 3 þ j4Þe j5t
Exercise 1.8
Express the following harmonic motion in the four equivalent forms:
h i xðx, tÞ ¼ Re ð 4 þ j3Þe j45 e jð2t 5xÞ
(Answers): (You must show all your work for full credit!)
(Form 2) |
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(Form 1) |
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(Form 4) |
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cos (2t |
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(Form 3) |
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1.9References of Trigonometric Identities
A collection of trigonometric identities is shown in this section for ease of reference. The derivations of most of these formulas are not difficult but are prone to small mistakes. The goal of this collection is to increase confidence and reduce mistakes when working with trigonometric functions and complex exponential functions for harmonic motion.
1.9.1Trigonometric Identities of a Single Angle
Trigonometric identities of a single angle can be found by the unit circle:
22 1 Complex Numbers for Harmonic Functions
cos ðθÞ ¼ cos ð θÞ sin ðθÞ ¼ sin ð θÞ
Some identities of trigonometric functions can be found by playing around with the unit circle:
sin ðθÞ ¼ sin ðθ πÞ ¼ sin ðθ þ πÞ
cos ðθÞ ¼ cos ðθ πÞ ¼ cos ðθ þ πÞ
The sine of any angle is equal to the cosine of its complementary angle. The relationship between the sine and cosine of complementary angles is:
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cos ðθÞ ¼ sin 2 |
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Euler’s formula:
e jθ ¼ cos ðθÞ j sin ðθÞ
or:
ejθ ¼ cos ðθÞ þ j sin ðθÞ; e jθ ¼ cos ðθÞ j sin ðθÞ
The reverse of Euler’s formula:
cos ðθÞ ¼ 12 ejθ þ e jθ ¼ Re ejθ
sin ðθÞ ¼ 21j ejθ e jθ ¼ 12 jejθ je jθ ¼ Re jejθ
Sine and cosine functions can be expressed in complex exponential functions as:
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