Roots of sin(z)=a

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Question statement

Find the roots of the equation $\sin(z)=\var{a1}$ , in the form $u+iv$ , where $u$ and $v$ are real, but $u$ depends on some index $n=0,\pm 1,\pm 2,\ldots$ , say.

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			Name	Type	Generated Value

a1	integer	2
v1	number	1.317
v2	number	-1.317
tol	number	0.001

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3. Gap Gap 2. Number entry
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$u=$ Gap 0.. (Do not enter decimals in your answer. Use $n$ for the index, and if you need to enter $\pi$ , write pi.)

There are two possible numerical answers $v_1$ and $v_2$ for the imaginary part. Enter the larger value in the first box, and the smaller value in the second box, both to 3d.p.

$v_1=$ Gap 1.. (Enter your answer to 3d.p.)

$v_2=$ Gap 2.. (Enter your answer to 3d.p.)

Advice

The equation $\sin(z)=\var{a1}$ can be solved by making the substitution $z=x+iy$ , and using the identity

$\sin(x+iy)=\sin(x)\cosh(y)+i\cos(x)\sinh(y),$

so then

$\sin(x)\cosh(y)+i\cos(x)\sinh(y)=\var{a1}.$

Now equate real and imaginary parts, so

$\begin{align}\sin(x)\cosh(y)&=\var{a1},\tag{1}\\\cos(x)\sinh(y)&=0.\tag{2}\end{align}$

From equation (2), either $\cos(x)=0$ or $\sinh(y)=0$ .

If $\sinh(y)=0$ , then $y=0$ , which would imply $\sin(x)=\var{a1}$ from equation (1). This is impossible, however, because $-1\leqslant\sin(x)\leqslant 1$ . We must have $\cos(x)=0$ , therefore, and so

$x=\frac{(1+2n)\pi}{2},\quad n=0,\pm 1,\pm 2,\ldots.$

Substituting this value of $x$ into equation (1) implies that $\pm\cosh(y)=\var{a1}$ , because $\sin(x)=\pm 1$ .

Because $\cosh(y)>0\;\forall y$ , however, $\sin(x)=-1$ is not permitted. This occurs for $n=\pm 1,\pm 3,\ldots$ . These values of $x$ are not permitted, therefore, so we must have

$x=\frac{(1+4n)\pi}{2},\quad n=0,\pm 1,\pm 2,\ldots.$

then $\cosh(y)=\var{a1}$ , and therefore

$y=\operatorname{arcosh}(\var{a1})\;\text{or}\;y=-\operatorname{arcosh}(\var{a1}).$

The roots are then $u+iv$ , with

$u=\frac{(1+4n)\pi}{2},\quad n=0,\pm 1,\pm 2,\ldots$

and

$v_{1,2}=\pm\operatorname{arcosh}(\var{a1})=\pm\var{v1}\;\text{to 3d.p.}$

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Complex variables by Newcastle University Mathematics and Statistics in Content created by Newcastle University.

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Roots of sin(z)=a

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: integer

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Gap-fill

Generated value: `integer`