Solving Trig Problem: sin^2000(x) + cos^2000(x)=1 - Kirstin

In summary, the conversation is about solving the trigonometry problem sin^2000(x) + cos^2000(x) = 1, where the solution of any multiple of pi is mentioned. The conversation also discusses the use of substitution and concludes that there are no other solutions.
  • #1
kirstin.17
5
0
I'm trying to solve this trig problem:

sin^2000(x) + cos^2000(x) = 1

I'm not sure how to go about it... I tried starting with sin^2(x) + cos^2(x) = 1 and build up to 2000 but I didn't get very far.

Obviously any multiple of pi will be an answer since either sin^2000 or cos^2000 will be 1 and the other will be 0. Are there others as well?

thx
-Kirstin.
 
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  • #2
You KNOW that [itex]\sin^{2}(x)+\cos^{2}(x)=1[/tex]
Set [itex]a=\sin^{1998}(x), b=\cos^{1998}(x)[/itex]
and your equation can be written as:
[tex]a\sin^{2}(x)+b\cos^{2}(x)=1[/tex]

Subtract the first from the second, yielding:
[tex](a-1)\sin^{2}(x)+(b-1)\cos^{2}(x)=0[/tex]
What can you conclude about this expression?
 
  • #3
Aha... I would say there are no other solutions other than what I already mentioned.

First note that the last expression you stated is equivalent to the original one we want to solve.

Solutions will occur either where [tex](a-1)\sin^{2}(x)=0[/tex] and [tex](b-1)\cos^{2}(x) =0[/tex], or where [tex](a-1)\sin^{2}(x)=-(b-1)\cos^{2}(x)[/tex].

The first situation has the solutions mentioned earlier.

The second situation does not have any solutions since LHS is always negative (since 0 < a < 1) and RHS is always positive.
 
  • #4
Indeed you are correct. :approve:
 
  • #5
......
good ans
 

1. What is the purpose of solving trigonometry problems?

The purpose of solving trigonometry problems is to find the values of unknown angles or sides in a triangle, using various trigonometric functions such as sine, cosine, and tangent.

2. How do you solve a trigonometry problem with the given equation sin^2000(x) + cos^2000(x)=1?

To solve this equation, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1. By substituting this into the given equation, we get sin^2000(x) + cos^2000(x) = sin^2(x) + cos^2(x) = 1. Therefore, the solution is any value of x that satisfies sin^2(x) + cos^2(x) = 1.

3. How do you determine the solutions for a trigonometry problem?

The solutions for a trigonometry problem can be determined by using trigonometric identities, such as the Pythagorean identity or double angle formulas, and solving for the unknown variable.

4. Can you explain the concept of degrees and radians in solving trigonometry problems?

Degrees and radians are two units of measurement used in trigonometry. Degrees measure angles in terms of 360 equal parts, while radians measure angles in terms of the radius of a circle. In order to use trigonometric functions, we need to convert between degrees and radians depending on the problem given.

5. What are some common mistakes to avoid when solving trigonometry problems?

Some common mistakes to avoid when solving trigonometry problems include forgetting to convert between degrees and radians, using the wrong trigonometric function for a given problem, and not double checking the solutions to ensure they are within the appropriate range for the specific problem.

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