91Ó°ÊÓ

Understanding trigonometric identities is essential when dealing with trigonometric functions. These identities are mathematical equations that hold true for all values of the variable. They serve as tools to simplify expressions, solve trigonometric equations, and transform functions into more manageable forms.

For instance, one fundamental identity is the Pythagorean identity, which states that for any angle \( x \), \( \sin^2{x} + \cos^2{x} = 1 \). This can be helpful when needing to find one trigonometric function value when another is known. Other commonly used identities include angle sum and difference formulas, and double and half-angle formulas.

When transforming the function \(f(x) = 3 \sin x + 4 \cos x + 10\), we rely on the identity for sine of a sum, which is \( \sin(a+b) = \sin a \cos b + \cos a \sin b \). This allows the expression to be rewritten in a form involving \( R \sin(x-a) + C \) by finding suitable values for \( R \) and \( a \).

Transformation of trigonometric expressions is about re-writing these expressions into equivalent forms that might be more advantageous for a certain purpose, such as solving equations or finding maxima and minima.

In our given problem, the expression \( 3 \sin x + 4 \cos x \) does not easily reveal its maximum and minimum values. To find these extremal values, we transform the expression into \( R \sin(x - a) + C \) where \( R \) is a positive real number, and \( a \) is an angle such that \( -\pi/2 \leq a \leq \pi/2 \).

The coefficients are used to determine \( R \) which represents the amplitude of the function and is calculated by \( \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the coefficients of \( \sin x \) and \( \cos x \) respectively. Through this transformation, the function becomes much easier to analyze since the maxima and minima of \( \sin \) are well-known and equal to 1 and -1.

Finding extrema, which are the maximum or minimum values of a function, in calculus typically involves using the first and second derivative tests. However, for simple trigonometric functions of the form \( R \sin(x - a) + C \), finding extrema can be done by analyzing the function directly.

Since we know the sine function oscillates between -1 and 1, any function of the form \( R \sin \theta \) will have its maxima and minima at \( R \) and \( -R \) respectively. Adding a constant \( C \) shifts the entire function upward by \( C \) units without changing the oscillation amplitude.

In our exercise, once the function \(f(x) = 3 \sin x + 4 \cos x + 10\) is transformed into the form \( 5 \sin(x + a) + 10 \) where \( a = \tan^{-1}(4/3) \), it is straightforward to determine that the maximum of \(f(x)\) occurs at \( 5 \cdot 1 + 10 = 15 \) and the minimum at \( 5 \cdot -1 + 10 = 5 \) without invoking calculus.