Find the principal value of the following expressions.

Ответ:

sammysosa121832

27.01.2022

Mathematics

(a) -0.266255 -0.963903i

(b) -1.381098

Step-by-step explanation:

A suitable calculator is very handy for such questions. See attached.

Euler's relation and the definitions of sin and cosh in exponential terms are helpful.

(a)

$(-1)^{\sqrt{2}}=e^{i\pi\sqrt{2}}=\cos{(\sqrt{2}\cdot180^{\circ})}+i\sin{(\sqrt{2}\cdot180^{\circ})}\\\\\approx\cos{254.558^{\circ}}+i\sin{254.558^{\circ}}\\\\\approx\boxed{-0.26625534-0.96390253i}$

(b)

$\sin^2{i}=(-i\sinh{(-1)})^2=-\sinh^2{(-1)}\approx\boxed{-1.38109785}$

(c)

$\cosh{(\sqrt{2}+3i)}=\dfrac{e^{\sqrt{2}+3i}+e^{-(\sqrt{2}+3i)}}{2}\\\\=\dfrac{e^{\sqrt{2}}(\cos{3}+i\sin{3})+e^{-\sqrt{2}}(\cos{(-3)}+i\sin{(-3)})}{2}\\\\=\cos{(3)}\cosh{\sqrt{2}}+i\sin{(3)}\sinh{\sqrt{2}}\\\\=\boxed{-2.15638538+0.27307665i}$

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The useful relations are ...

$e^{i\theta}=\cos{\theta}+i\sin{\theta}\\\\\sin(\theta)=\dfrac{e^{i\theta}-e^{-i\theta}}{2i}=-i\sinh{(i\theta)}\\\\\cosh{(\theta)}=\dfrac{e^{\theta}+e^{-\theta}}{2},\ \sinh{(\theta)}=\dfrac{e^{\theta}-e^{-\theta}}{2}$

Find the principal value of the following expressions. Write your answer in the form a + ib.(a)(b) (

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Ответ:

sophiaa23

24.03.2022

Mathematics

GIVE BRAINLIEST

If we take a trapezoid and split the height into 2 parts, flip the top part around, and connect it to the side of the bottom part, we get a parallelogram. This is how the formula to find a trapezoid works:

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Find the principal value of the following expressions. Write your answer in the form a + ib.

(a)

(b)

(c)

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Find the principal value of the following expressions. Write your answer in the form a + ib.(a)(b) (c)

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Find the principal value of the following expressions. Write your answer in the form a + ib.

(a)

(b)

(c)