Can cos^2(kt)+sin^2(kt) always equal 1, regardless of the value of k?

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Discussion Overview

The discussion revolves around the mathematical identity cos²(kt) + sin²(kt) and whether it holds true for any constant value of k. Participants explore the implications of this identity in relation to trigonometric functions and their geometric interpretations.

Discussion Character

  • Technical explanation

Main Points Raised

  • One participant notes that k is a constant and expresses difficulty in finding verification for the identity.
  • Another participant asserts that cos²(θ) + sin²(θ) = 1 for any value of θ, including θ = kt, referencing the Pythagorean identity and its geometric basis on the unit circle.
  • A later reply expresses gratitude for the explanation and connects it to a broader application regarding force vectors for alloys.
  • Another participant shares personal satisfaction in resolving a long-standing question related to the identity.

Areas of Agreement / Disagreement

Participants generally agree on the validity of the identity cos²(kt) + sin²(kt) = 1 for any constant k, but the initial question about verification remains somewhat unresolved.

Contextual Notes

The discussion does not address potential limitations or assumptions regarding the application of the identity in different contexts.

mill
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Where k is a constant.

I am trying to simplify a problem but the t has constant in front of it that is not one, and I can't seem to find a chart that verifies it either way.
 
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Yes. cos2θ + sin2θ = 1 for any value of θ, including θ = kt. This is called the Pythagorean identity, and is the result of applying the Pythagorean theorem to the fact that cosθ and sinθ are the x and y coordinates, respectively, of a point on the unit circle (a circle of radius 1).
 
Thanks!
 
Thanks!

slider142 said:
Yes. cos2θ + sin2θ = 1 for any value of θ, including θ = kt. This is called the Pythagorean identity, and is the result of applying the Pythagorean theorem to the fact that cosθ and sinθ are the x and y coordinates, respectively, of a point on the unit circle (a circle of radius 1).
...is a brilliant answer to my problem of how to describe force vectors for alloys. Big thanks, slider142. and mill.
 
Just got back from seaside and found answer to a month's headache. Complementary! B)
 

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