Can this trigonometric equation be solved for x_1 and x_2 in terms of \alpha?

In summary: There is no general method to solve this equation, but there are methods that can be used depending on the situation.
  • #1
physicsRookie
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[tex]x_1(cos\alpha-1) + x_2sin\alpha = 0 [/tex]
[tex]x_1sin\alpha + x_2(-cos\alpha-1) = 0 [/tex]
How to solve this equation? Can anyone help me?
 
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  • #2
It's a system of equations: 2 equations in 2 unknowns. That means you can solve it. Just solve for one unknown in terms of the other using the first equation, and then subsitute that into the second.
 
  • #3
Let me try...

Solve equation 1:
[tex]x_2 = \frac{-x_1(cos\alpha - 1)}{sin\alpha}[/tex]

Substitute it to the second:
[tex]x_1sin\alpha + \frac{-x_1(cos\alpha - 1)}{sin\alpha}(-cos\alpha-1) = 0[/tex]

[tex]x_1sin\alpha + \frac{x_1(cos^2\alpha - 1)}{sin\alpha} = 0[/tex]

[tex]2x_1sin\alpha = 0[/tex]

What is the solutions for [tex]2x_1sin\alpha = 0[/tex]?
Obviously one is [tex]x_1=0[/tex], but if [tex]sin\alpha = 0[/tex], then...
 
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  • #4
physicsRookie said:
Let me try...

Solve equation 1:
[tex]x_2 = \frac{-x_1(cos\alpha - 1)}{sin\alpha}[/tex]

Substitute it to the second:
[tex]x_1sin\alpha + \frac{-x_1(cos\alpha - 1)}{sin\alpha}(-cos\alpha-1) = 0[/tex]

[tex]x_1sin\alpha + \frac{x_1(cos^2\alpha - 1)}{sin\alpha} = 0[/tex]

[tex]2x_1sin\alpha = 0[/tex]

What is the solutions for [tex]2x_1sin\alpha = 0[/tex]?
Obviously one is [tex]x_1=0[/tex], but if [tex]sin\alpha = 0[/tex], then...
Well done. The point is, of course, that [tex]\alpha[/tex] is a number (not one of the variables) so these can be solved like any pair of equations for x1 and x2.
Notice, by the way, that if [tex]sin\alpha= 0[/tex], your first step, dividing by that, would be invalid. You have to look at this case separately.
If [tex]sin\alpha= 0[/tex], then [tex]cos\alpha[/tex] is either 1 or -1.

What do your equations look like if [tex]sin\alpha= 0[/tex] and [tex]cos\alpha= 1[/tex]?

What do your equations look like if [tex]sin\alpha= 0[/tex] and [tex]cos\alpha= -1[/tex]?
 
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  • #5
HallsofIvy, thanks.
HallsofIvy said:
What do your equations look like if [tex]sin\alpha= 0[/tex] and [tex]cos\alpha= 1[/tex]?
[tex]2x_1=0 and 0=0 => x_1=0, x_2 [/tex] could be any number
HallsofIvy said:
What do your equations look like if [tex]sin\alpha= 0[/tex] and [tex]cos\alpha= -1[/tex]?
[tex] 0=0 and -2x_2=0 => x_2=0, x_1 [/tex] could be any number

I just try another solution.
Rewrite the equations:
[tex](x_1cos\alpha + x_2sin\alpha) - x_1 = 0 [/tex]
[tex](x_1sin\alpha - x_2cos\alpha) - x_2 = 0 [/tex]

Suppose [tex]x_1 = cos\frac{\alpha}{2}, x_2 = sin\frac{\alpha}{2}[/tex], then

[tex]cos\frac{\alpha}{2}cos\alpha + sin\frac{\alpha}{2}sin\alpha - cos\frac{\alpha}{2} = 0 [/tex]

[tex]cos\frac{\alpha}{2}sin\alpha - sin\frac{\alpha}{2}cos\alpha - sin\frac{\alpha}{2} = 0 [/tex]

It works!

I am wondering whether there is some general method to solve [tex]x_1, x_2[/tex] depending on [tex]\alpha[/tex] or not.
 
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1. What is a trigonometric equation?

A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent, and one or more variables. The solution to a trigonometric equation is the value of the variable that satisfies the equation.

2. How do I solve a trigonometric equation?

To solve a trigonometric equation, you can use algebraic manipulation and trigonometric identities to rewrite the equation in terms of a single trigonometric function. Then, you can use inverse trigonometric functions to find the solution(s) for the variable.

3. What are common trigonometric identities?

Some common trigonometric identities include the Pythagorean identities, double angle identities, and sum and difference identities. These identities can be used to simplify trigonometric expressions and solve trigonometric equations.

4. How do I check my solution to a trigonometric equation?

You can check your solution to a trigonometric equation by substituting the value of the variable into the original equation and verifying that it satisfies the equation. You can also use a graphing calculator to graph the equation and the solution point to visually confirm the solution.

5. Are there any special cases to consider when solving trigonometric equations?

Yes, there are some special cases to consider when solving trigonometric equations. For example, some equations may have no solution, infinite solutions, or solutions restricted to a certain interval. It is important to carefully check the domain and range of the trigonometric functions involved in the equation to determine these special cases.

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