1. ## CIS problems

A small portion of the last chapter in my textbook is devoted to cis problems. It is very unclear about the cis trigometric function, providing no examples whatsoever.

Here are two problems from my textbook I attempted to work on:

"Write in a + bi form: $5 cis 300°$
Round the value of a to the hundredth.

and

"Multiply: $(6 cis 100°) (2 cis 70°)$

If I could get an explanation on these two problems, I'll (hoprefully) be able to work though the rest, and I'll have a good study session. I'm guaranteed at least one question on cis on my final.

At first I thought the folks who wrote the book made a typo for cos, given the "I" key on the keyboard is so close to the "O" key.

Many thanks in advance to quick help!

2. Originally Posted by wiseguy
A small portion of the last chapter in my textbook is devoted to cis problems. It is very unclear about the cis trigometric function, providing no examples whatsoever.

Here are two problems from my textbook I attempted to work on:

"Write in a + bi form: $5 cis 300°$
Round the value of a to the hundredth.

and

"Multiply: $(6 cis 100°) (2 cis 70°)$

If I could get an explanation on these two problems, I'll (hoprefully) be able to work though the rest, and I'll have a good study session. I'm guaranteed at least one question on cis on my final.

At first I thought the folks who wrote the book made a typo for cos, given the "I" key on the keyboard is so close to the "O" key.

Many thanks in advance to quick help!

You have to (1) understand that $cis(\theta):=\cos\theta+i\sin\theta$ , (2) you have to know pretty well your trigonometry and (3) you have to remember the definition $a+ib=\sqrt{a^2+b^2}\,cis\theta\,,\,\,\theta=\arcta n\frac{b}{a}\,,\,\,a,b\in\mathbb{R}\,,\,a\neq 0$ , and if $a=0\,,\,\,\theta=\frac{\pi}{2}+k\pi\,,\,\,k\in\mat hbb{Z}$.

So let's do one, and you do the other one: $5\,cis(300^\circ)=5(\cos 300^\circ+i\sin 300^\circ)=5\left(\frac{1}{2}-\frac{\sqrt{3}}{2}i\right) = \frac{5}{2}-\frac{5\sqrt{3}}{2}i$

Tonio

3. Originally Posted by wiseguy
A small portion of the last chapter in my textbook is devoted to cis problems. It is very unclear about the cis trigometric function, providing no examples whatsoever.

Here are two problems from my textbook I attempted to work on:

"Write in a + bi form: $5 cis 300°$
Round the value of a to the hundredth.

and

"Multiply: $(6 cis 100°) (2 cis 70°)$

If I could get an explanation on these two problems, I'll (hoprefully) be able to work though the rest, and I'll have a good study session. I'm guaranteed at least one question on cis on my final.

At first I thought the folks who wrote the book made a typo for cos, given the "I" key on the keyboard is so close to the "O" key.

Many thanks in advance to quick help!
Hi Wiseguy...

"cis" means "cos+isin"

so $cis(\theta)=cos\theta+isin(\theta)$

For a complex number a+bi..."a" is the real part and "bi" is the imaginary part.

$5cis300^o=5\left(cos300^o+isin300^o\right)$

hence ... $a=5cos300^o$ and $b=5sin300^o$

$6cis100^o2cis670^o=6\left(cos100^o+isin100^o\right )2\left(cos70^o+isin70^o\right)$

$=12\left(cos100^ocos70^o+icos100^osin70^o+icos70^o sin100^o-sin100^osin70^o\right)$

Now combine the real parts to get "a" and the imaginary parts to get "b"
after applying the appropriate trigonometric identities to them.

4. Originally Posted by tonio
You have to (1) understand that $cis(\theta):=\cos\theta+i\sin\theta$ , (2) you have to know pretty well your trigonometry and (3) you have to remember the definition $a+ib=\sqrt{a^2+b^2}\,cis\theta\,,\,\,\theta=\arcta n\frac{b}{a}\,,\,\,a,b\in\mathbb{R}\,,\,a\neq 0$ , and if $a=0\,,\,\,\theta=\frac{\pi}{2}+k\pi\,,\,\,k\in\mat hbb{Z}$.

So let's do one, and you do the other one: $5\,cis(300^\circ)=5(\cos 300^\circ+i\sin 300^\circ)=5\left(\frac{1}{2}-\frac{\sqrt{3}}{2}i\right) = \frac{5}{2}-\frac{5\sqrt{3}}{2}i$

Tonio
Okay, I ended up with 12cis(-0.984807753)°
...this is the form they were asking for in the second question: "Leave your answer in the form ___cis___°"

5. Originally Posted by wiseguy
Okay, I ended up with 12cis(-0.984807753)°
...this is the form they were asking for in the second question: "Leave your answer in the form ___cis___°"
From $12\left(cos100^ocos70^o-sin100^osin70^o+i[cos100^osin70^o+cos70^osin100^o]\right)$

you should use $cosAcosB-sinAsinB=cos(A+B)$

and $cosAsinB+cosBsinA=sin(A+B)$

where $A=100^0$ and $B=70^o$

to get $12cis(A+B)$

6. Which equals 12cis170°...?

7. Originally Posted by wiseguy
Which equals 12cis170°...?
Yes,

you evaluated $cos170^o=-0.9848$

However $cis170^0=cos170^0+isin170^o$