Trigonometric Equations - Why Do I Always Get cos^3 or sin^3?

In summary, the conversation is about solving two similar equations involving cubic trigonometric functions. The equations are given in image form and the speaker is stuck on how to proceed with solving them. Another person suggests using trigonometric identities to change the equations into cubic equations in either cosine or sine. The speaker also mentions not knowing how to solve cubic trigonometric equations and the other person suggests using the rational root theorem to reduce the number of possible solutions. The conversation ends with the suggestion to start by checking simple values and then dividing the polynomial by a root to reduce it to a quadratic equation.
  • #1
okh
16
0

Homework Statement


I'm trying to solve two similar equations, but I can't go on.
This is the first one
http://img818.imageshack.us/img818/298/imageevsu.jpg

This is the second one:
http://img7.imageshack.us/img7/7812/imageowud.jpg

Homework Equations





The Attempt at a Solution


For the first one:
http://img855.imageshack.us/img855/4633/imagekhd.jpg

And for the second equation:
http://img12.imageshack.us/img12/2990/imageiag.jpg

And I'm stuck with these cos^3x and sin^3x. I think I should replace something...
 
Last edited by a moderator:
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  • #2
okh said:

Homework Statement


I'm trying to solve two similar equations, but I can't go on.
This is the first one
http://img818.imageshack.us/img818/298/imageevsu.jpg

This is the second one:
http://img7.imageshack.us/img7/7812/imageowud.jpg

Homework Equations



The Attempt at a Solution


For the first one:
http://img855.imageshack.us/img855/4633/imagekhd.jpg

And for the second equation:
http://img12.imageshack.us/img12/2990/imageiag.jpg

And I'm stuck with these cos^3x and sin^3x. I think I should replace something...
For the first one, change sin2(x) to 1-cos2(x). You will have a cubic equation in cos(x).

For the second one, change cos2(x) to 1-sin2(x). You will have a cubic equation in sin(x).
 
Last edited by a moderator:
  • #3
Okay, but how to solve cubic trigonometic equations?
I think I can get a quadratic equation, in some way. We are studying them, indeed, and there is nothing about cubic trigonometric equations in my book.
 
  • #4
Well, if you don't do something you can't possibly get an answer!

Have you tried just checking some simple values? It helps sometimes to know the "rational root theorem". Any rational root, of the form p/q with integers p and q, of the polynoial [itex]a_nx^n+ a_{n-1}x^{n-1}+ a_1x+ a_0= 0[/itex] must have p evenly dividing [itex]a_0[/itex] and q evenly dividing [itex]a_n[/itex]. That will reduce the number of trials. Of course, it is not necessary that a polynomial equation have rational roots but this one does.

Once you have a single root, say x= a, divide the cubic polynomial by x- a to reduce to a quadratic.
 

Related to Trigonometric Equations - Why Do I Always Get cos^3 or sin^3?

1. Why do I always get cos^3 or sin^3 when solving trigonometric equations?

Trigonometric equations often involve the use of trigonometric identities, such as the power-reducing identities for cosine and sine. These identities state that cos^2x = (1 + cos2x)/2 and sin^2x = (1 - cos2x)/2. When solving for a variable in a trigonometric equation, these identities can result in cos^3 or sin^3 appearing in the solution.

2. Can I avoid getting cos^3 or sin^3 in my solutions?

In some cases, it is possible to manipulate the equation using other identities or trigonometric functions to avoid getting cos^3 or sin^3 in the solution. However, in many cases, these terms will appear due to the nature of trigonometric equations.

3. Are cos^3 and sin^3 interchangeable?

No, cos^3 and sin^3 are not interchangeable. They represent different trigonometric functions and cannot be used in place of one another in an equation. It is important to use the correct trigonometric function based on the given equation or problem.

4. Can I simplify cos^3 or sin^3 in my solution?

Yes, cos^3 and sin^3 can be simplified using trigonometric identities. For example, cos^3x can be rewritten as cosx(cos^2x), and sin^3x can be rewritten as sinx(sin^2x). These expressions can then be simplified further using the power-reducing identities mentioned earlier.

5. Are there any strategies for effectively solving equations with cos^3 or sin^3?

One strategy for solving equations with cos^3 or sin^3 is to use the power-reducing identities to rewrite them as a combination of cos^2x and sin^2x. From there, the equation can be solved as a quadratic equation, which may be easier to solve. Another strategy is to use a graphing calculator or software to graph the equation and find the points of intersection, which represent the solutions to the equation.

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