Saurabh Koranglekar
Last Activity: 5 Years ago
To demonstrate that the equation possesses real solutions only when , 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 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 and 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 and is 1. Therefore, the maximum possible value of can be expressed as:
This is derived from the fact that can be rewritten using the angle sum identity. Specifically, we can express it in the form , where and is a phase shift. The maximum value occurs when , yielding .
Establishing the Condition for Real Solutions
For the equation to have real solutions, the value of must fall within the range defined by the maximum and minimum values of . Thus, we require:
- Minimum value:
- Maximum value:
This leads us to the conclusion that for real solutions to exist, must satisfy:
Deriving the Squared Condition
When we square the inequalities, we obtain:
- Left inequality:
- Right inequality is always satisfied, since squaring a negative value gives a positive result.
This shows that the condition is both necessary and sufficient for the existence of real solutions to the equation .
Final Thoughts
In summary, the equation can be solved for real if and only if . 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.