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From the topic of Trigonometric functions of Complex Numbers. Find the principal values of csc(3-7j).

From the topic of Trigonometric functions of Complex Numbers. Find the principal values of csc(3-7j).

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter4: Graphing And Inverse Functions
Section4.3: Vertical And Horizontal Translations
Problem 69PS
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From the topic of Trigonometric functions of Complex Numbers.

Find the principal values of csc(3-7j).

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) eix - 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) 2j sin jy = j sinh y (6) This is guide only 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 + yi) = 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) Solution: tan(3 – 7j) = sin(3-7)) 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 3-7) tan(3 – 7j) = sin cos(3-7)) 77.37+542.83/ (542.83+77.37j) = -i
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) eix - 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) 2j sin jy = j sinh y (6) This is guide only 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 + yi) = 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) Solution: tan(3 – 7j) = sin(3-7)) 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 3-7) tan(3 – 7j) = sin cos(3-7)) 77.37+542.83/ (542.83+77.37j) = -i
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