Why is the range of asinx+bcosx is same as range of asinx+bcosx??

Your question touches on a fascinating aspect of trigonometric functions and their transformations. The expression you provided seems to contain a small typo, but I assume you're asking why the range of the function \( a \sin x + b \cos x \) is the same as the range of \( a \sin x + b \cos x \). Let’s clarify this concept step by step.

Understanding the Function

The general form \( a \sin x + b \cos x \) represents a linear combination of sine and cosine functions. The coefficients \( a \) and \( b \) determine the amplitude and phase of the resulting wave. The key point here is that both sine and cosine functions oscillate between -1 and 1, which influences the combined function's behavior.

Finding the Range

To find the range of \( a \sin x + b \cos x \), we can use a geometric interpretation. The expression can be rewritten in a form that allows us to analyze its maximum and minimum values:

• We can express \( a \sin x + b \cos x \) as \( R \sin(x + \phi) \), where:
• \( R = \sqrt{a^2 + b^2} \), which represents the resultant amplitude.
• \( \phi \) is a phase shift determined by the coefficients \( a \) and \( b \).
• This transformation utilizes the identity for sine of a sum: \( R \sin(x + \phi) = R(\sin x \cos \phi + \cos x \sin \phi) \).

The maximum value of \( R \sin(x + \phi) \) is \( R \) (when \( \sin(x + \phi) = 1 \)), and the minimum value is \( -R \) (when \( \sin(x + \phi) = -1 \)). Therefore, the range of \( a \sin x + b \cos x \) is given by:

Range Calculation

Thus, the range is:

• From \( -\sqrt{a^2 + b^2} \) to \( \sqrt{a^2 + b^2} \).

This means that regardless of how you express the function, the range remains the same, as it depends solely on the values of \( a \) and \( b \), and not on the specific form of the expression.

Real-World Analogy

Think of this in terms of physical movement, like a pendulum swinging. The height it can reach (maximum and minimum) depends on the length of the pendulum (analogous to \( R \)), irrespective of the angle at which it swings (analogous to the phase shift \( \phi \)). The total swing range remains constant as long as the length (or the values of \( a \) and \( b \)) does not change.

Conclusion

In summary, both expressions \( a \sin x + b \cos x \) reflect the same underlying oscillatory behavior governed by their coefficients. The maximum and minimum values derived from their combined amplitude dictate the range, making it consistent across different representations of the same function.