# [Homework Help] Find the value of z when sec(z) = 3i.

• Questioneer
In summary, to find the value of z when sec(z) = 3i, we can use the fact that sec(z) can be rewritten as 1/cos(z) and then use the complex form of cos(z) to obtain a quadratic equation.
Questioneer

## Homework Statement

Find the value of z when sec(z) = 3i.

## Homework Equations

sec (z) = 1/cos(z)

## The Attempt at a Solution

I assume that I need to transform into polar coordinates and then use some transformations. I'm really at a loss. If not a solution maybe a few hints to go in the right direction?

Questioneer said:

## Homework Statement

Find the value of z when sec(z) = 3i.

## Homework Equations

sec (z) = 1/cos(z)
Good start. And, since you are working with complex numbers, cos(z)= (eiz+ e-iz)/2. So you have
[tex]\frac{2}{e^z+ e^{-z}}= i[/itex]
which can be reduced to
$e^{iz}+ e^{-iz}= -2i$
and that can be converted to a quadratic.

## The Attempt at a Solution

I assume that I need to transform into polar coordinates and then use some transformations. I'm really at a loss. If not a solution maybe a few hints to go in the right direction?
Don't assume things!

## 1. What is the definition of sec(z)?

Sec(z) is the reciprocal of cosine, or 1/cos(z). It represents the ratio of the hypotenuse to the adjacent side of a right triangle in trigonometry.

## 2. How do you find the value of z when sec(z) = 3i?

To find the value of z, you can use the inverse cosine function. In this case, you would take the inverse cosine of 1/3i, which is equivalent to the inverse cosine of -3i/3. This gives you an angle of 1.230959 radians, or approximately 70.53 degrees. However, keep in mind that there are multiple possible solutions for z in this case, as secant has a period of 2π.

## 3. Can the value of z be negative?

Yes, the value of z can be negative. In fact, in this case, z could also have a value of -1.230959 radians or -70.53 degrees, as well as any other angle that is an odd multiple of π/2 away from these values.

## 4. How does the value of z change if the value of sec(z) changes?

As sec(z) represents the ratio of the hypotenuse to the adjacent side of a right triangle, the value of z will change as the ratio changes. If sec(z) decreases, the value of z will increase, and vice versa.

## 5. Can the value of z be imaginary?

Technically, yes, the value of z can be imaginary. However, in this case, the value of z would be considered a complex number, as it would have both a real and imaginary component. In the context of trigonometry, it is more common for z to be a real number, as it represents an angle.

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