Whats a non-trivial linear combination of these functions?

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Discussion Overview

The discussion revolves around finding a non-trivial linear combination of the functions f(x)=17, g(x)=2Sin²(x), and h(x)=3Cos²(x) that vanishes identically. Participants explore the relationships between these functions and seek to determine the constants C1, C2, and C3 that satisfy the equation C1f + C2g + C3h = 0.

Discussion Character

  • Mathematical reasoning
  • Exploratory
  • Homework-related

Main Points Raised

  • One participant suggests that C1 must be 0 due to the lack of a constant relation between the constant function and the trigonometric functions.
  • Another participant reminds that the identity sin²(x) + cos²(x) = 1 could be relevant to the problem.
  • A participant attempts to manipulate the equation C2Sin²(x) = -C3Cos²(x) using trigonometric identities but does not arrive at a constant solution.
  • There is a suggestion to express 17 in terms of sin²(x) and cos²(x) to facilitate finding the coefficients.
  • Participants discuss applying coefficients to the trigonometric functions to achieve a balance with the constant 17.

Areas of Agreement / Disagreement

Participants express various approaches to the problem, but no consensus is reached on the specific values of C2 and C3 or the method to derive them. The discussion remains unresolved regarding the exact linear combination that satisfies the condition.

Contextual Notes

Participants note the challenge of simplifying the relationships between the functions and the constants, indicating potential limitations in their current approaches.

warfreak131
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I have to find a non-trivial, linear combination of the following functions that vanishes identically.

In other words

C1f + C2g + C3h = 0

Where C1, C2, and C3 are all constant, and cannot all = 0.

f(x)=17
g(x)=2Sin2(x)
h(x)=3Cos2(x)

I figure C1 = 0, because there's really no constant relation between the trig functions and 17.

That means that C22Sin2(x)=-C33Cos2(x)

I need help finding C2 and C3. I've already tried substituting with trig identities, but I am getting nothing as of now.
 
Last edited:
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hi warfreak131! :smile:

c'mon … think! … what's the most famous equation involving cos2 and sin2 ? :wink:
 
tiny-tim said:
hi warfreak131! :smile:

c'mon … think! … what's the most famous equation involving cos2 and sin2 ? :wink:

I know that sin2x + cos2x = 1, and I've tried that, but I am still not getting it

C22Sin2(x)=-C33Cos2(x)

C22(1-cos2(x))=-C33Cos2(x)

C2(2-2cos2(x))=-C33cos2(x)

but this doesn't simplify into a constant solution, as far as i worked it out
 
warfreak131 said:
I know that sin2x + cos2x = 1 …

ok, so what = 17? :smile:

get some sleep! :zzz:​
 
tiny-tim said:
ok, so what = 17? :smile:

get some sleep! :zzz:​
17sin2x+17cos2x?

so C1 = 17Sin2x + 17Cos2x

then just apply another coefficient to 2Sin2x, and 3Cos2x to make the new coefficient equal 17?
 
Last edited:
yup! :smile:

g'night! :zzz:​
 
awesome thanks
 

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