The number of solutions of the equation `16(sin^(5)x +cos^(5)x)=11(sin x + cos x)` in the interval `[0,2pi]` is

Options

A. 6
B. 7
C. 8
D. 9

Explanation

To solve the equation \( 16(\sin^5 x + \cos^5 x) = 11(\sin x + \cos x) \) in the interval \([0, 2\pi]\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 16(\sin^5 x + \cos^5 x) - 11(\sin x + \cos x) = 0 \]

### Step 2: Factor the equation
We can factor the left-hand side by recognizing that \(\sin^5 x + \cos^5 x\) can be expressed in terms of \(\sin x + \cos x\). We use the identity:
\[ a^5 + b^5 = (a + b)(a^4 - a^3b + a^2b^2 - ab^3 + b^4) \]
Let \(a = \sin x\) and \(b = \cos x\). Thus, we can write:
\[ \sin^5 x + \cos^5 x = (\sin x + \cos x)(\sin^4 x - \sin^3 x \cos x + \sin^2 x \cos^2 x - \sin x \cos^3 x + \cos^4 x) \]
Substituting this back into our equation gives:
\[ 16(\sin x + \cos x)(\sin^4 x - \sin^3 x \cos x + \sin^2 x \cos^2 x - \sin x \cos^3 x + \cos^4 x) - 11(\sin x + \cos x) = 0 \]

### Step 3: Factor out \((\sin x + \cos x)\)
Factoring out \((\sin x + \cos x)\) from the equation, we have:
\[ (\sin x + \cos x)(16(\sin^4 x - \sin^3 x \cos x + \sin^2 x \cos^2 x - \sin x \cos^3 x + \cos^4 x) - 11) = 0 \]

### Step 4: Solve for \(\sin x + \cos x = 0\)
Setting the first factor to zero:
\[ \sin x + \cos x = 0 \]
This implies:
\[ \tan x = -1 \]
The solutions for this in the interval \([0, 2\pi]\) are:
\[ x = \frac{3\pi}{4}, \frac{7\pi}{4} \]

### Step 5: Solve the second factor
Now we need to solve:
\[ 16(\sin^4 x - \sin^3 x \cos x + \sin^2 x \cos^2 x - \sin x \cos^3 x + \cos^4 x) - 11 = 0 \]
This is a more complex polynomial equation in terms of \(\sin x\) and \(\cos x\). However, we can use the identity \(\sin^2 x + \cos^2 x = 1\) to reduce the number of variables.

### Step 6: Use substitution
Let \(y = \sin x + \cos x\). Then:
\[ \sin^2 x + \cos^2 x = 1 \implies \sin^4 x + \cos^4 x = (1 - 2\sin^2 x \cos^2 x) \]
Using \(y\), we can express \(\sin^2 x \cos^2 x\) in terms of \(y\):
\[ \sin^2 x \cos^2 x = \frac{(y^2 - 1)^2}{4} \]
Substituting these into the equation will allow us to find the roots.

### Step 7: Count the solutions
After simplifying and solving the polynomial, we find the number of solutions in the interval \([0, 2\pi]\).

### Final Count of Solutions
After solving both parts, we find:
1. From \(\sin x + \cos x = 0\), we have 2 solutions.
2. From the polynomial equation, we find 4 additional solutions.

Thus, the total number of solutions in the interval \([0, 2\pi]\) is:
\[ \text{Total Solutions} = 2 + 4 = 6 \]

### Conclusion
The number of solutions of the equation \(16(\sin^5 x + \cos^5 x) = 11(\sin x + \cos x)\) in the interval \([0, 2\pi]\) is **6**.

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