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Can anyone prove that asinx+bcosx=c has a possible solution in reals only if c^2 is les than equal to a^2+b^2?

Khushi goyal , 8 Years ago
Grade 11
anser 1 Answers
Saurabh Koranglekar

Last Activity: 5 Years ago

To demonstrate that the equation asinx+bcosx=c possesses real solutions only when c2a2+b2, we can utilize some fundamental properties of trigonometric functions and vector analysis. Let's break this down step by step.

Understanding the Equation

The expression asinx+bcosx can be viewed as a linear combination of sine and cosine functions. This can be interpreted geometrically as a projection of a point onto a line in a two-dimensional plane, where a and b represent the coefficients that determine the amplitude of the sine and cosine components, respectively.

Using the Pythagorean Identity

We know from trigonometric identities that the maximum value of sinx and cosx is 1. Therefore, the maximum possible value of asinx+bcosx can be expressed as:

  • Maximum value: a2+b2

This is derived from the fact that asinx+bcosx can be rewritten using the angle sum identity. Specifically, we can express it in the form Rsin(x+ϕ), where R=a2+b2 and ϕ is a phase shift. The maximum value occurs when sin(x+ϕ)=1, yielding R.

Establishing the Condition for Real Solutions

For the equation asinx+bcosx=c to have real solutions, the value of c must fall within the range defined by the maximum and minimum values of asinx+bcosx. Thus, we require:

  • Minimum value: a2+b2
  • Maximum value: a2+b2

This leads us to the conclusion that for real solutions to exist, c must satisfy:

  • Condition: a2+b2ca2+b2

Deriving the Squared Condition

When we square the inequalities, we obtain:

  • Left inequality: c2a2+b2
  • Right inequality is always satisfied, since squaring a negative value gives a positive result.

This shows that the condition c2a2+b2 is both necessary and sufficient for the existence of real solutions to the equation asinx+bcosx=c.

Final Thoughts

In summary, the equation asinx+bcosx=c can be solved for real x if and only if c2a2+b2. This relationship beautifully illustrates how trigonometric functions can be linked to geometric interpretations, reinforcing the idea that the interplay between algebra and geometry is fundamental in mathematics.

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