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 \( 5 \sec \theta - 13 = 12 \tan \theta \) 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:
\[ 5 \cdot \frac{1}{\cos \theta} - 13 = 12 \cdot \frac{\sin \theta}{\cos \theta} \]
Multiplying through by \(\cos \theta\) (assuming \(\cos \theta \neq 0\)):
\[ 5 - 13 \cos \theta = 12 \sin \theta \]

### Step 2: Rearranging the equation
Rearranging the equation, we have:
\[ 5 - 12 \sin \theta = 13 \cos \theta \]

### Step 3: Square both sides
Squaring both sides to eliminate the square root gives:
\[ (5 - 12 \sin \theta)^2 = (13 \cos \theta)^2 \]
Expanding both sides:
\[ 25 - 120 \sin \theta + 144 \sin^2 \theta = 169 \cos^2 \theta \]

### Step 4: 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 5: Combine like terms
Rearranging gives:
\[ 144 \sin^2 \theta + 169 \sin^2 \theta - 120 \sin \theta + 25 - 169 = 0 \]
This simplifies to:
\[ 313 \sin^2 \theta - 120 \sin \theta - 144 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \( \sin \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ \sin \theta = \frac{120 \pm \sqrt{(-120)^2 - 4 \cdot 313 \cdot (-144)}}{2 \cdot 313} \]
Calculating the discriminant:
\[ \sin \theta = \frac{120 \pm \sqrt{14400 + 180288}}{626} \]
\[ \sin \theta = \frac{120 \pm \sqrt{194688}}{626} \]
Calculating the square root gives:
\[ \sin \theta = \frac{120 \pm 441.23}{626} \]

### Step 7: Find possible values for \(\sin \theta\)
Calculating the two possible values:
1. \( \sin \theta = \frac{561.23}{626} \approx 0.897 \)
2. \( \sin \theta = \frac{-321.23}{626} \approx -0.513 \)

### Step 8: Determine the angles
For \( \sin \theta \approx 0.897 \):
- The angles are in the first and second quadrants:
- \( \theta_1 = \sin^{-1}(0.897) \)
- \( \theta_2 = \pi - \sin^{-1}(0.897) \)

For \( \sin \theta \approx -0.513 \):
- The angles are in the third and fourth quadrants:
- \( \theta_3 = 2\pi + \sin^{-1}(-0.513) \)
- \( \theta_4 = \pi - \sin^{-1}(-0.513) \)

### Step 9: Count the solutions
Thus, we have a total of 4 solutions in the interval \([0, 2\pi]\).

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