The number of integral values of 'k' for which the equation 3 sin x + 4 cos x = k + 1 has a solution

Q: Wait, why can't we just solve for $x$ directly?

That is a great question! If you try to solve for $x$ directly, you will find yourself stuck because $x$ is not fixed; it is a variable. The equation asks for the values of $k$ that allow *at least one* $x$ to satisfy the equation. This is a range problem, not an algebraic solving problem. We need to find the range of the function $f(x) = 3 \sin x + 4 \cos x$ and see if the horizontal line $y = k + 1$ intersects it.

Q: Why is the range of $a \sin x + b \cos x$ always $[-\sqrt{a^2+b^2}, \sqrt{a^2+b^2}]$?

This comes from the harmonic addition theorem. You can rewrite $a \sin x + b \cos x$ as $R \sin(x + \phi)$, where $R = \sqrt{a^2+b^2}$ and $\tan \phi = b/a$. Since the sine function $\sin(x + \phi)$ always oscillates between $-1$ and $1$, the entire expression must oscillate between $-R$ and $R$. It is a beautiful geometric property of waves!

Q: Why do we include the endpoints $-6$ and $4$ in our count?

The inequality we derived is $-6 \le k \le 4$. The question asks for the number of integral values of $k$ for which the equation has a solution. If $k = -6$, then $k + 1 = -5$, which is the minimum value of the wave. Since the wave reaches $-5$, the equation $3 \sin x + 4 \cos x = -5$ has a solution. The same logic applies to $k = 4$, where $k + 1 = 5$, the maximum value. Therefore, the endpoints are valid solutions.

Question

The number of integral values of '$k$' for which the equation $3 \sin x + 4 \cos x = k + 1$ has a solution, $k \in R$ is

Sigma Percentile · Accepted Answer

### Analyzing the Wave Envelope

Imagine you are standing before a calm lake, watching a single ripple move across the surface. In mathematics, the expression $3 \sin x + 4 \cos x$ is exactly like that ripple—a wave that rises and falls with rhythmic predictability.

Many students approach this problem by trying to isolate $x$, but that is like trying to catch the wind. Instead, we must look at the 'envelope' of the wave—the boundaries it can never cross.

### The Anatomy of the Wave

Our equation is $3 \sin x + 4 \cos x = k + 1$. The left-hand side is a classic superposition of two waves.

In the world of trigonometry, any expression of the form $a \sin x + b \cos x$ can be compressed into a single sine wave with an amplitude of $R = \sqrt{a^2 + b^2}$. Here, our coefficients are $a = 3$ and $b = 4$.

When we calculate the amplitude, we find:

$$R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

This means our wave is perfectly trapped between the horizontal lines $y = -5$ and $y = 5$. It can never reach $6$, and it can never drop to $-7$. It is bound by the laws of geometry.

### The Geometry of Existence

Now, consider the right-hand side: $k + 1$. This is not a wave; it is a horizontal line.

For our equation to have a solution, this line must intersect our wave. If the line $y = k + 1$ is floating above $5$ or sinking below $-5$, the wave and the line will never meet, and the equation will have no solution.

Therefore, for a solution to exist, the value $k + 1$ must be trapped within the same boundaries as our wave:

$$-5 \le k + 1 \le 5$$

### The Final Inequality

This is the moment of truth. We have the inequality $-5 \le k + 1 \le 5$. To find the values of $k$, we simply subtract $1$ from every part of the inequality.

This gives us:

$$-6 \le k \le 4$$

Now, we just need to count the integers in this range. We have the set $\{-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4\}$.

If you count them carefully, you will find there are exactly **11** values.

By understanding the geometric soul of the trigonometric function, we turned a potentially confusing algebraic problem into a simple counting exercise. Whenever you see a complex equation, ask yourself: "What is the range of this function?" The answer often reveals the path to the solution.

Animated Solution for Mathematics - Trigonometry: The number of integral values of ' $k$ ' for which the equation $3 sin x + 4 cos x = k + 1$ has a solution, $k \in R$ is

Enter Numerical Value:

Visualized Solution

Equation Analysis

Range of $a sin x + b cos x$

Identify Coefficients

Calculate $a^{2} + b^{2}$

Find the Square Root

Define the Range

Condition for Solution

Set up Inequality

Isolate $k$

List Integral Values

Final Count

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Wave Envelope

The Anatomy of the Wave

The Geometry of Existence

The Final Inequality

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Animated Solution for Mathematics - Trigonometry: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of integral values of 'k' for which the equation 3sinx+4cosx=k+1 has a solution, k∈R is