# Ugly trig and exponential problem

• gulfcoastfella
In summary, the student is trying different methods to solve an equation that is nonlinear and beyond their ability. They have been unsuccessful thus far.
gulfcoastfella
Gold Member

## Homework Statement

This isn't a problem, I'm just obsessed with analyzing a trig/exp equation algebraically instead of with a calculator.

8*sin(t) - 16*cos(t) = 9*exp(-t/2)

## Homework Equations

See part (1.) above...

## The Attempt at a Solution

I tried converting the exp part into a Fourier series; if you graph the equation, this method recommends itself due to the multiple solutions. I didn't have much success with this method, though, since the first term in the Fourier series contains two exp terms. In addition, the Fourier series requires the definition of an integrable "period", and I wouldn't know what to do for that.

I also tried converting the sine and cosine terms into exp terms with complex arguments, but then I've got imaginary numbers all over the place, and I already know the answer isn't complex from the solution obtained by calculator, which was verified graphically.

Any recommendations would be grand.

gulfcoastfella said:

## Homework Statement

This isn't a problem, I'm just obsessed with analyzing a trig/exp equation algebraically instead of with a calculator.

8*sin(t) - 16*cos(t) = 9*exp(-t/2)

## Homework Equations

See part (1.) above...

## The Attempt at a Solution

I tried converting the exp part into a Fourier series; if you graph the equation, this method recommends itself due to the multiple solutions. I didn't have much success with this method, though, since the first term in the Fourier series contains two exp terms. In addition, the Fourier series requires the definition of an integrable "period", and I wouldn't know what to do for that.

I also tried converting the sine and cosine terms into exp terms with complex arguments, but then I've got imaginary numbers all over the place, and I already know the answer isn't complex from the solution obtained by calculator, which was verified graphically.

Any recommendations would be grand.

I forgot to mention another method I attempted...
I converted both sides of the equation (the trig side and the exp side) to Taylor series. The terms line up beautifully, but the constants multiplied on each term in the original equation throw the equations out of whack.

I've decided that this equation is so far past non-linear that it can't be solved algebraically. I'm going to go with the graphical intersection method, and let the problem go.

Did you try writing sin(x) as $(e^{ix}- e^{-ix})/(2i)$ and cos(x) as $(e^{ix}+ e^{-ix})/2$ so everything is in terms of the exponential?

HallsofIvy said:
Did you try writing sin(x) as $(e^{ix}- e^{-ix})/(2i)$ and cos(x) as $(e^{ix}+ e^{-ix})/2$ so everything is in terms of the exponential?

I tried that, but couldn't figure out where to go with it.

There is no algebraic solution to this.

Avodyne said:
There is no algebraic solution to this.

As I suspected... is there a reason or proof or reference that you can link me to?

Thanks.

## 1. What is a "trig" function?

A trigonometric function, or "trig" function, is a mathematical function that relates the angles of a triangle to the lengths of its sides. Examples of trig functions include sine, cosine, and tangent.

## 2. What is an "exponential" function?

An exponential function is a mathematical function in which a constant, known as the base, is raised to a variable power. It is commonly written as f(x) = ab^x, where a is the initial value and b is the base.

## 3. How are trig and exponential functions related?

Trig functions and exponential functions are related because they both involve the use of a variable raised to a power. In fact, the sine and cosine functions can be expressed in terms of exponential functions using Euler's formula.

## 4. Why are "ugly" trig and exponential problems difficult to solve?

"Ugly" trig and exponential problems are difficult to solve because they often involve complicated equations and multiple steps. They may also require the use of advanced mathematical techniques, making them challenging for many students.

## 5. How can I improve my understanding of "ugly" trig and exponential problems?

To improve your understanding of "ugly" trig and exponential problems, it is important to practice solving a variety of problems and to seek help from a teacher or tutor when needed. Additionally, familiarizing yourself with common trig and exponential identities and rules can make solving these problems easier.

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