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Find the principal value, show your full Solutions. Evaluate sec−1(3-j).

Find the principal value, show your full Solutions. Evaluate sec−1(3-j).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.3: The Addition And Subtraction Formulas
Problem 71E
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Find the principal value, show your full Solutions.

Evaluate sec−1(3-j).

3.2 Inverse Trigonometric Functions Example 3 Evaluate sin -1(1 + 2j) Solution: Let x = sin -1(1+ 2j) sin x = 1+2j eix-e-jx = 1+ 2j elx – e-jx = (1 + 2j)(2j) eix - e-jx = -4 + 2/ Multiply both sides by elx (ejx) – 1 = (-4 + 2j)e}* (eix)“ + eix (4 – 2j) – 1 = 0 elx - -(4-2j)±y(4-2j)2-4(1)(-1) 2(1) -4+2/+v162-161 eix = 2 V162 – 16i = v22.627 L - 22.5° = 4.395 – 1.82i eix =4+2j±(4.395-1.82i) 2 ejx = 0.1975 + 0,09i and eix = -4.1975 + 1.91i jx = In(0.1975 + 0.091) = In(0.217 e0.4276j) jx = In(0.217) + 0.4276j jx = -1.528 + 0.4276j x =. 4276 + 1.528j Then other value: elx = -4.1975 + 1.91/ jx = In(-4.1975 + 1.91j) = In(4.612e2.7146) = In 4.612 + 2.7146j jx = 1.529 + 2.7146j x = 2.7146 – 1. 528j Example 4 Evaluate tan-1(3 + 4j) Solution: let x = tan-(y) where y = 3 + 4j tan x = y [(ejx-e-jx)][(eix-e-jx = y 2j eix - e-jx = j(eix – e-jx)y Multiply both sides by elx (el*)° – 1 = j(e/*)° + 1)y (el*)°(1 – y) = 1 + jy (eix)? = 1+/y 1-jy Substitute y = 3+ 4j and use calculator (e*)², = -0.7059 + 0.1765 eix = (0.728 e/2.897)/2 = 0.853 e1.4485j jx = In (0.853 e1.4485)) = In 0.853 + 1.4485j jx = -0.159 + 1.4485j x = 1.4485 + 0.159 j The other value is: eix = -(0.853 e1.4485j) = 0.10406 – 0.847j = 0.853e-1.693j Jx = In (0.853e-1.693/ ) = In 0.853 – 1.693j = -0.159 – 1.693j Then: x = -1.693 + 0.159j
Transcribed Image Text:3.2 Inverse Trigonometric Functions Example 3 Evaluate sin -1(1 + 2j) Solution: Let x = sin -1(1+ 2j) sin x = 1+2j eix-e-jx = 1+ 2j elx – e-jx = (1 + 2j)(2j) eix - e-jx = -4 + 2/ Multiply both sides by elx (ejx) – 1 = (-4 + 2j)e}* (eix)“ + eix (4 – 2j) – 1 = 0 elx - -(4-2j)±y(4-2j)2-4(1)(-1) 2(1) -4+2/+v162-161 eix = 2 V162 – 16i = v22.627 L - 22.5° = 4.395 – 1.82i eix =4+2j±(4.395-1.82i) 2 ejx = 0.1975 + 0,09i and eix = -4.1975 + 1.91i jx = In(0.1975 + 0.091) = In(0.217 e0.4276j) jx = In(0.217) + 0.4276j jx = -1.528 + 0.4276j x =. 4276 + 1.528j Then other value: elx = -4.1975 + 1.91/ jx = In(-4.1975 + 1.91j) = In(4.612e2.7146) = In 4.612 + 2.7146j jx = 1.529 + 2.7146j x = 2.7146 – 1. 528j Example 4 Evaluate tan-1(3 + 4j) Solution: let x = tan-(y) where y = 3 + 4j tan x = y [(ejx-e-jx)][(eix-e-jx = y 2j eix - e-jx = j(eix – e-jx)y Multiply both sides by elx (el*)° – 1 = j(e/*)° + 1)y (el*)°(1 – y) = 1 + jy (eix)? = 1+/y 1-jy Substitute y = 3+ 4j and use calculator (e*)², = -0.7059 + 0.1765 eix = (0.728 e/2.897)/2 = 0.853 e1.4485j jx = In (0.853 e1.4485)) = In 0.853 + 1.4485j jx = -0.159 + 1.4485j x = 1.4485 + 0.159 j The other value is: eix = -(0.853 e1.4485j) = 0.10406 – 0.847j = 0.853e-1.693j Jx = In (0.853e-1.693/ ) = In 0.853 – 1.693j = -0.159 – 1.693j Then: x = -1.693 + 0.159j
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