If \(3 \cos x+4 \sin x=K\) has a possible solution then number of values of integral \(\mathrm{K}\) is (a) 3 (b) 5 (c) 10 (d) 11

Understanding the sine addition formula is crucial to solving a wide range of trigonometric problems. This potent formula allows us to combine the sine of two angles in a simplified manner. Specifically, the formula is defined as \( \sin(a + b) = \sin a\cos b + \cos a\sin b \).

When you encounter an equation like \(3 \cos x + 4 \sin x = K\), it might not be immediately clear how it relates to the sine addition formula. However, with some clever manipulation, this unwieldy expression can be expressed in the form of \(\sin(a + b)\), making it solvable. By identifying \(a = x\), and finding comparable \(\sin b\) and \(\cos b\) values, we can rewrite the equation in the recognizable pattern of the sine addition formula. This method transforms a complex trigonometric equation into something far more manageable, showcasing the sheer utility of this identity in solving equations.

The world of trigonometry is underpinned by an array of identities that provide relationships between different trigonometric functions. These trigonometric identities are essential tools for simplifying expressions and solving equations. An identity is an equation that holds true for any value of the variable where the functions are defined.

In the context of the given equation, \(3 \cos x + 4 \sin x = K\), we've applied a trigonometric identity to transform it into a format that uses the sine addition formula. Another common set of identities are the Pythagorean identities, like \(\sin^2 x + \cos^2 x = 1\), which can also prove incredibly useful in various scenarios. Recognizing when and how to apply these identities enables the successful simplification and resolution of trigonometric equations. Furthermore, they facilitate a deeper understanding of the underlying relationships between trigonometric functions.

Each trigonometric function has specific ranges that they adhere to, and knowing these ranges is pivotal when solving trigonometric equations. The range is the set of all possible values a function can output. For instance, the range of the sine and cosine functions is \( [-1,1] \), which means their outputs vary between -1 and 1, inclusive.

In our example, we use the range of the sine function to determine the range of possible values for \(K\) when given \(3 \cos x + 4 \sin x = K\). We already deduced that \(\sin(a + b)\) is constrained within the bounds of -1 and 1. Consequently, multiplying this range by 5, we understand that \(K\) can be any value from -5 to 5. This step is vital to solve for the number of integral values that \(K\) can take within its range. Grasping the concept of ranges not only assists in solving such equations but also helps in predicting the behavior of trigonometric functions across different intervals.