If ` sec^(2) theta + tan^(2) theta = (5)/(3) and 0 le theta le (pi)/(2)` then the value of ` theta` is equal to

Options

A. `15^(@)`
B. ` 30^(@)`
C. ` 45^(@)`
D. ` 60^(@)`

Answer

To solve the equation \( \sec^2 \theta + \tan^2 \theta = \frac{5}{3} \) for \( \theta \) in the range \( 0 \leq \theta \leq \frac{\pi}{2} \), we can follow these steps:

### Step 1: Use the identity
We know the trigonometric identity:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
Using this identity, we can rewrite the equation.

### Step 2: Substitute the identity into the equation
Substituting \( \sec^2 \theta \) in the original equation:
\[ 1 + \tan^2 \theta + \tan^2 \theta = \frac{5}{3} \]
This simplifies to:
\[ 1 + 2\tan^2 \theta = \frac{5}{3} \]

### Step 3: Isolate the term with \( \tan^2 \theta \)
Subtract 1 from both sides:
\[ 2\tan^2 \theta = \frac{5}{3} - 1 \]
Converting 1 to a fraction with a denominator of 3:
\[ 2\tan^2 \theta = \frac{5}{3} - \frac{3}{3} = \frac{2}{3} \]

### Step 4: Solve for \( \tan^2 \theta \)
Now, divide both sides by 2:
\[ \tan^2 \theta = \frac{2}{3} \cdot \frac{1}{2} = \frac{1}{3} \]

### Step 5: Find \( \tan \theta \)
Taking the square root of both sides:
\[ \tan \theta = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]

### Step 6: Determine the angle \( \theta \)
We know that \( \tan 30^\circ = \frac{1}{\sqrt{3}} \). Therefore:
\[ \theta = 30^\circ \]

### Conclusion
Thus, the value of \( \theta \) is:
\[ \theta = 30^\circ \quad \text{or} \quad \theta = \frac{\pi}{6} \text{ radians} \]

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