The number of solution of the equation ` 5 sec theta -13=12 tan theta `in `[0,2pi]` is

Options

A. 2
B. 1
C. 4
D. 0

Answer

To solve the equation \( 5 \sec \theta - 13 = 12 \tan \theta \) for the number of solutions in the interval \([0, 2\pi]\), we will follow these steps:

### Step 1: Rewrite the equation in terms of sine and cosine
We know that:
\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Substituting these into the equation gives:
\[ 5 \cdot \frac{1}{\cos \theta} - 13 = 12 \cdot \frac{\sin \theta}{\cos \theta} \]

### Step 2: Multiply through by \(\cos \theta\) to eliminate the denominators
Assuming \(\cos \theta \neq 0\), we can multiply through by \(\cos \theta\):
\[ 5 - 13 \cos \theta = 12 \sin \theta \]

### Step 3: Rearrange the equation
Rearranging gives:
\[ 5 - 12 \sin \theta = 13 \cos \theta \]

### Step 4: Square both sides
To eliminate the sine and cosine, we will square both sides:
\[ (5 - 12 \sin \theta)^2 = (13 \cos \theta)^2 \]
Expanding both sides:
\[ 25 - 120 \sin \theta + 144 \sin^2 \theta = 169 \cos^2 \theta \]

### Step 5: Use the Pythagorean identity
Using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\):
\[ 25 - 120 \sin \theta + 144 \sin^2 \theta = 169(1 - \sin^2 \theta) \]
Expanding the right side:
\[ 25 - 120 \sin \theta + 144 \sin^2 \theta = 169 - 169 \sin^2 \theta \]

### Step 6: Combine like terms
Bringing all terms to one side:
\[ 144 \sin^2 \theta + 169 \sin^2 \theta - 120 \sin \theta + 25 - 169 = 0 \]
This simplifies to:
\[ 313 \sin^2 \theta - 120 \sin \theta - 144 = 0 \]

### Step 7: Solve the quadratic equation
Let \( t = \sin \theta \). The quadratic equation is:
\[ 313t^2 - 120t - 144 = 0 \]
Using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{120 \pm \sqrt{(-120)^2 - 4 \cdot 313 \cdot (-144)}}{2 \cdot 313} \]
Calculating the discriminant:
\[ b^2 - 4ac = 14400 + 180288 = 194688 \]
Thus, we have:
\[ t = \frac{120 \pm \sqrt{194688}}{626} \]
Calculating the square root:
\[ \sqrt{194688} \approx 441.23 \]
So:
\[ t_1 = \frac{120 + 441.23}{626} \approx 0.89, \quad t_2 = \frac{120 - 441.23}{626} \approx -0.51 \]

### Step 8: Determine valid solutions for \( \sin \theta \)
The valid solution for \( \sin \theta \) must be in the range \([-1, 1]\):
- \( \sin \theta = 0.89 \) is valid.
- \( \sin \theta = -0.51 \) is valid.

### Step 9: Find angles corresponding to these sine values
For \( \sin \theta = 0.89 \):
- \( \theta_1 = \sin^{-1}(0.89) \)
- \( \theta_2 = \pi - \sin^{-1}(0.89) \)

For \( \sin \theta = -0.51 \):
- \( \theta_3 = \sin^{-1}(-0.51) \)
- \( \theta_4 = 2\pi + \sin^{-1}(-0.51) \)

### Step 10: Count the solutions in the interval \([0, 2\pi]\)
Each sine value gives two angles in the interval \([0, 2\pi]\):
- From \( \sin \theta = 0.89 \), we get 2 solutions.
- From \( \sin \theta = -0.51 \), we also get 2 solutions.

Thus, the total number of solutions is:
\[ \text{Total solutions} = 2 + 2 = 4 \]

### Final Answer
The number of solutions of the equation \( 5 \sec \theta - 13 = 12 \tan \theta \) in the interval \([0, 2\pi]\) is **4**.

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