Trig. form of a Complex No.?

Question

If z=1+i
then what is its trigonometric form?
And how to get the argument (i.e. Tita 0)???

Yes, if this is the ans 1/root(2){cos 45 + i sin 45} then how do you get 45 deg?

Thanks a ton math_kp! You solved my Q totally!
And yes, Hy you are right you have to express it in radians!

Thank you all!

Answer 1

in trigonomertic form

z = r cos t + ir sin t = 1 + i

so r cos t = 1
r sin t = 1

square and add r^2 = 2
so r = sqrt(2)

now cos t = 1/sqrt(2) and sint = 1/sqrt(2)

so t = pi/4 (as cos pi/4 = 1/sqrt(2) and sin is also positive so 1st quadrant)

so t = pi/4 and r = sqrt(2)

Answer 2

There's an awful lot of blah-blah going on here.

Look, in general, when z=a+bi then tanθ=b/a, so
here tanθ=1/1, hence θ=45 degrees or more properly, π/4.
The length L is b/sin(π/4) or a/cos(π/4) so here we have
L=1/sin(π/4)=1/(1/2^(1/2) = 1.414 or 2^(1/2).

You could also use the Pythagorean theorem and say
L=(1^2 + 1^2)^(1/2) = 2^(1/2).

On the complex plane this is L{cos(π/4) + (i)sin(π/4)} and

in polar form it's (L,(π/4)).

Now here's a hint. In math we tend to use the same angles a
lot. They are the 45d, 30d and 60d. In radians : π/4, π/6 and π/3
respectively. In a right triangle the ratio of the sides is side to side to hypotenuse as follows:
for a 45d(π/4) - 1:1:2^(1/2)
for a 30,60 (π/3, π/6) - 1:2:3^(1/2) .
There's also the 3:4:5 rt triangle and occasionally the 5:12:13.

Get familiar with these ratios, you can usually get around
the trig with them..

Answer 3

On an Argand diagram, the point (x,y)
represents the complex number z = x + iy.

Your complex number is z = 1 + i, or 1 + i1,
which means x = 1 and y = 1.

Now, x = r*cos(theta) and y = r*sin(theta)
and r = sqrt(x^2 + y^2) = sqrt(1 + 1) = sqrt(2).

But if x = y = 1, then r*cos(theta) = r*sin(theta)

Dividing both sides by r*cos(theta) gives :

1 = tan(theta). Therefore, theta = pi / 4 rads.

The polar form is : z = x + iy = r[cos(theta) + i*sin(theta)]

Therefore, z = 1 + i1 = sqrt(2) * [cos(pi / 4) + i*sin(pi / 4)]

Answer 4

You mean its the polar form?
Then it would be 1/root(2){cos 45 + i sin 45}

Answer 5

For z = x + iy,
tan(arg) = y/x

I believe it's wrong to use degrees for expressing complex numbers in this form. In this case
arctan(y/x) = arctan 1
= pi/4

PS Yeah, I agree, Math KP should get the points.

Answer 6

z = 1 + 1i
z = V2 {(1/V2) + (i/V2) }
z = V2{cos45 + i sin45}. The angle in degrees.

Th