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?

To demonstrate that the equation \( a \sin x + b \cos x = c \) possesses real solutions only when \( c^2 \leq a^2 + b^2 \), 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 \( a \sin x + b \cos x \) 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 \( \sin x \) and \( \cos x \) is 1. Therefore, the maximum possible value of \( a \sin x + b \cos x \) can be expressed as:

• Maximum value: \( \sqrt{a^2 + b^2} \)

This is derived from the fact that \( a \sin x + b \cos x \) can be rewritten using the angle sum identity. Specifically, we can express it in the form \( R \sin(x + \phi) \), where \( R = \sqrt{a^2 + b^2} \) and \( \phi \) is a phase shift. The maximum value occurs when \( \sin(x + \phi) = 1 \), yielding \( R \).

Establishing the Condition for Real Solutions

For the equation \( a \sin x + b \cos x = c \) to have real solutions, the value of \( c \) must fall within the range defined by the maximum and minimum values of \( a \sin x + b \cos x \). Thus, we require:

• Minimum value: \( -\sqrt{a^2 + b^2} \)
• Maximum value: \( \sqrt{a^2 + b^2} \)

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

• Condition: \( -\sqrt{a^2 + b^2} \leq c \leq \sqrt{a^2 + b^2} \)

Deriving the Squared Condition

When we square the inequalities, we obtain:

• Left inequality: \( c^2 \leq a^2 + b^2 \)
• Right inequality is always satisfied, since squaring a negative value gives a positive result.

This shows that the condition \( c^2 \leq a^2 + b^2 \) is both necessary and sufficient for the existence of real solutions to the equation \( a \sin x + b \cos x = c \).

Final Thoughts

In summary, the equation \( a \sin x + b \cos x = c \) can be solved for real \( x \) if and only if \( c^2 \leq a^2 + b^2 \). 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.