If `0^@ le theta le 90^@`, find the value of `theta` satisfying 3 tan `theta + cot theta =5` cosec `theta`

Answer

To solve the equation \(3 \tan \theta + \cot \theta = 5 \csc \theta\) for \(\theta\) in the range \(0^\circ \leq \theta \leq 90^\circ\), we will follow these steps:

### Step 1: Rewrite the equation in terms of sine and cosine
We know the following trigonometric identities:
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
- \(\csc \theta = \frac{1}{\sin \theta}\)

Substituting these identities into the equation gives:
\[ 3 \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = 5 \frac{1}{\sin \theta} \]

### Step 2: Multiply through by \(\sin \theta \cos \theta\) to eliminate the fractions
Multiplying both sides by \(\sin \theta \cos \theta\) results in:
\[ 3 \sin^2 \theta + \cos^2 \theta = 5 \cos \theta \]

### Step 3: Use the Pythagorean identity
We know that \(\sin^2 \theta + \cos^2 \theta = 1\). Therefore, we can express \(\sin^2 \theta\) as \(1 - \cos^2 \theta\):
\[ 3(1 - \cos^2 \theta) + \cos^2 \theta = 5 \cos \theta \]

### Step 4: Simplify the equation
Expanding this gives:
\[ 3 - 3 \cos^2 \theta + \cos^2 \theta = 5 \cos \theta \]
Combining like terms results in:
\[ 3 - 2 \cos^2 \theta = 5 \cos \theta \]

### Step 5: Rearrange to form a quadratic equation
Rearranging the equation gives:
\[ 2 \cos^2 \theta + 5 \cos \theta - 3 = 0 \]

### Step 6: Solve the quadratic equation
Let \(x = \cos \theta\). The equation becomes:
\[ 2x^2 + 5x - 3 = 0 \]
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
- Here, \(a = 2\), \(b = 5\), and \(c = -3\).
- The discriminant is \(b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-3) = 25 + 24 = 49\).

Calculating the roots:
\[ x = \frac{-5 \pm \sqrt{49}}{2 \cdot 2} = \frac{-5 \pm 7}{4} \]

This gives us two possible solutions:
1. \(x = \frac{2}{4} = \frac{1}{2}\)
2. \(x = \frac{-12}{4} = -3\) (not valid since \(\cos \theta\) cannot be negative in the first quadrant)

### Step 7: Find \(\theta\)
Since \(\cos \theta = \frac{1}{2}\), we find:
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

### Final Answer
Thus, the value of \(\theta\) satisfying the equation is:
\[ \theta = 60^\circ \]