Efficient Method for Finding Constants in Trig Function Power Problem

In summary, the conversation discusses finding constants a, b, and c in order to solve the equation sin(∅)^5 = asin∅ + bsin3∅ + csin5∅. The suggested methods for solving this include using the exponential form of sin and expanding it using the binomial theorem, or using the identity for sin(A+B) to write sin3∅ and sin5∅ in terms of sin∅ and cos∅, and then using the identity cos^2∅ = 1- sin^2∅ to write the equation as a polynomial in sin∅.
  • #1
Pandabasher
10
0

Homework Statement


Find constants a, b and c such that
sin(∅)^5=asin∅+bsin3∅+csin5∅



Homework Equations





The Attempt at a Solution


I expressed sin in its complex form (if that's what its called) and put it to the power 5, and then multiplied each bracket out one at a time, and got the correct answers, as checked by wolfram, but I was wondering if there is a quicker way to find the bracket to the power 5 other than just multiplying each bracket out 5 times with its self? This is probably a simple question but I never seem to be able to get the simple stuff. Any help is greatly appreciated. Thanks.

Sorry for the wording; I don't know how to use latex, and it's really ugly just typed out.
 
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  • #2
Using the exponential form of sin is probably the fastest way to work this out. Expanding [itex](e^{i\phi} - e^{-i\phi})^5[/itex] might be a bit simpler using the binomial theorem: http://en.wikipedia.org/wiki/Binomial_theorem

An alternative to using complex numbers is to use the identity for [itex]\sin(A+B)[/itex] to write [itex]\sin 3\phi[/itex] and [itex]\sin 5\phi[/itex] in terms of [itex]\sin\phi[/itex] and [itex]\cos\phi[/itex]. Then use [itex]\cos^2\phi = 1-\sin^2\phi[/itex] to write the whole thing as a polynomial in [itex]\sin\phi[/itex].
 
  • #3
Ah, thanks for clearing that up. My tutor used the binomial theorem, but didn't mention it, probably because he's been doing it for so long. Thanks for your help :)
 

1. What is a trigonometric function power problem?

A trigonometric function power problem is an equation or mathematical expression that involves a trigonometric function (such as sine, cosine, or tangent) raised to a power. These types of problems often require the use of trigonometric identities and properties to solve.

2. How do I solve a trigonometric function power problem?

To solve a trigonometric function power problem, you will typically need to use trigonometric identities and properties to simplify the expression and then use algebraic techniques to solve for the unknown variable. It may also be helpful to graph the equation to visualize the solution.

3. What are some common trigonometric identities and properties used in these types of problems?

Some common trigonometric identities and properties used in solving trigonometric function power problems include the Pythagorean identities, reciprocal identities, double angle identities, and sum and difference identities.

4. How can I check my solution to a trigonometric function power problem?

To check your solution to a trigonometric function power problem, you can substitute your answer back into the original equation and see if it satisfies the equation. You can also use a graphing calculator to graph both the original equation and your solution to visually confirm that they intersect at the same point.

5. What are some real-world applications of trigonometric function power problems?

Trigonometric function power problems can be used to solve a variety of real-world problems, such as calculating the height of a building or the distance between two points. They are also commonly used in fields such as engineering, physics, and astronomy to model and solve various phenomena and equations.

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