MCS 351 ENGINEERING MATHEMATICS SOLUTION OF ...

10. cos 3πi = ei(3πi) +e −i(3πi)
2
= e−3π +e 3π
2
= cosh 3π.
11. cosh(−2 + 3i) = e−2+3i +e 2−3i
2
= e−2 (cos 3+i sin 3)+e 2 (cos(−3)+isin(−3))
2
= cos 3( e−2 +e 2
2
) + i sin 3( e−2 −e 2
2
)
= cos 3 cosh 2 − i sin 3sinh2.
12. • −i sinh(−π + 2i) = −i( e−π+2i −e π−2i
2
)
• sin(2 + πi) =
= −i( e−π (cos 2+i sin 2)−e π (cos(−2)+i sin(−2))
2
)
= −i(cos 2( e−π −e π
2
) + i sin 2( e−π +e π
2
))
= −i(cos 2(− sinh π) + i sin 2 cosh π)
= sin 2 cosh π + i cos 2 sinh π.
sinh i(2+πi)
i
= sinh(−π+2i)
i
13. cosh(2n + 1)πi = e(2n+1)πi +e −(2n+1)πi
2
=
−i sinh(−π+2i)
1
= i cos 2 sinh π + sin 2 cosh π.
= e0 (cos((2n+1)π)+i sin((2n+1)π))+e 0 (cos((−2n−1)π)+i sin((−2n−1)π)
2
= cos((2n+1)π)+cos((−2n−1)π)
2
= cos(π)+cos(−π)
2
= −1, n = 1, 2, ...
14. sinh(4 − 3i) = e4−3i −e −4+3i
2
= e4 (cos(−3)+i sin(−3))−e −4 (cos 3+i sin 3)
2
= cos 3( e4 −e −4
2
) + i sin 3( −e4 −e −4
2
)
= cos 3 sinh 4 − i sin 3 cosh 4.
15. cosh(4 − 6πi) = e4−6πi −e −4+6πi
2
= e4 (cos(−6π)+i sin(−6π))+e −4 (cos(6π)+i sin(6π))
2
= e4 +e −4
2
= cosh 4.
16. We know that tan a = sin a
cos a
tan a+tan b
and tan(a + b) =
1−tan a tan b .
tan z = tan(x + iy) =
tan x+tan iy
1−tan x tan iy =
sin x sin iy
+ cos x cos iy
1− sin x
cos x
.
sin iy
cos iy
=
sin x cos iy+sin iy cos x
cos x cos iy
cos x cos iy−sin x sin iy
cos x cos iy
=
sin x cos iy+sin iy cos x
cos x cos iy−sin x sin iy
=
sin x cosh y+i sinh y cos x
cos x cosh y−i sin x sinh y = sin x cos x cosh2 y+i cos 2 x cosh y sinh y+i sin 2 x sinh y cosh y−sinh 2 y cos x sin x
cos 2 x cosh 2 y+sin 2 x sinh 2 y
= cosh y sinh y(i(cos2 x+sin 2 x))+cos x sin x(cosh 2 y−sinh 2 y)
cos 2 x(1+sinh 2 y)+sin 2 x sinh 2 y
=
i cosh y sinh y+cos x sin x
cos 2 x+cos 2 x sinh 2 y+sin 2 x sinh 2 y
=
cos x sin x+i cosh y sinh y
cos 2 x+sinh 2 y(cos 2 x+sin 2 x) =
Hence we get Re(tan z) =
17-21: Equations
cos x sin x cosh y sinh y
+ i
cos 2 x+sinh 2 y
cos x sin x
and Im(tan z) =
cos 2 x+sinh 2 y
17. 0 = cosh z = ez +e −z
2
= 1 2 (ez + 1
e
) = 1 e 2z +1
z 2 e
. z
cos 2 x+sinh 2 y
cosh y sinh y
cos 2 x+sinh 2 y .
21