Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Find the principal value.
Evaluate cot(-3+j).
![LESSON III.
Trigonometric Functions of Complex Numbers
3.1 Trigonometric Functions
From Euler's Formula:
eix = cos x +j sin x
e-jx = cos x - į sin x
(1)
(2)
Add (1) and (2)
elx + e-jx = 2 cos x
(ejx + e-ix)
cos x =
(3)
Subtract (1) and (2)
ejx - e-jx = 2jsin x
sin x =eR-e-jx)
2j
(4)
From (3) Let x = jy
e-y – ey = 2 cos(jy)
(e-y- e)
cos (jy) =
cos jy = cosh y
(5)
From (4) Let x = jy
sin(jy) = e-e)
2j
= / ()
sin jy = j sinh y
(6)
From 5and 6. We can now derive the trigonometric functions.
sin(x + jy) = sin x cos jy + cos x sin y
= sin x cosh y + jcos xsinh y
cos(x + yj) = cos x cosjy – sin x sin jy
= cos x coxh y - j sin xsinh y
sin(x+jy)
cos(x+jy)
tan(x + jy)
csc(x + jy) =
sin(x+jy)
sec(x + jy) =
cos(x+jy)
Example 2
Evaluate tan(3 – 7j)
sin(3-7))
Solution: tan(3 – 7j) =
cos(3-7/)
sin(3 – 7j) = sin 3 cosh(-7) + cos 3 sinh(-7)
= 0.1411(548.317) + (-0.99(-548.31)j
= 77.37 + 542.83j
cos(3 – 7j) = cos 3 cosh(-7) – j sinh(-7)
= 0.99(548.317) – j(0.1411)(-548.317)
= -542.83 + 77.37j
tan(3 – 7j) = sin3-7)
cos(3-7))
77.37+542.83/
(542.83+77.37j)
= -i](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe202a04f-cea3-4344-8030-185aa4de4707%2F38c91801-9020-4c26-8c92-20e5a271435a%2F0radip_processed.jpeg&w=3840&q=75)
Transcribed Image Text:LESSON III.
Trigonometric Functions of Complex Numbers
3.1 Trigonometric Functions
From Euler's Formula:
eix = cos x +j sin x
e-jx = cos x - į sin x
(1)
(2)
Add (1) and (2)
elx + e-jx = 2 cos x
(ejx + e-ix)
cos x =
(3)
Subtract (1) and (2)
ejx - e-jx = 2jsin x
sin x =eR-e-jx)
2j
(4)
From (3) Let x = jy
e-y – ey = 2 cos(jy)
(e-y- e)
cos (jy) =
cos jy = cosh y
(5)
From (4) Let x = jy
sin(jy) = e-e)
2j
= / ()
sin jy = j sinh y
(6)
From 5and 6. We can now derive the trigonometric functions.
sin(x + jy) = sin x cos jy + cos x sin y
= sin x cosh y + jcos xsinh y
cos(x + yj) = cos x cosjy – sin x sin jy
= cos x coxh y - j sin xsinh y
sin(x+jy)
cos(x+jy)
tan(x + jy)
csc(x + jy) =
sin(x+jy)
sec(x + jy) =
cos(x+jy)
Example 2
Evaluate tan(3 – 7j)
sin(3-7))
Solution: tan(3 – 7j) =
cos(3-7/)
sin(3 – 7j) = sin 3 cosh(-7) + cos 3 sinh(-7)
= 0.1411(548.317) + (-0.99(-548.31)j
= 77.37 + 542.83j
cos(3 – 7j) = cos 3 cosh(-7) – j sinh(-7)
= 0.99(548.317) – j(0.1411)(-548.317)
= -542.83 + 77.37j
tan(3 – 7j) = sin3-7)
cos(3-7))
77.37+542.83/
(542.83+77.37j)
= -i
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