If `0le theta le pi ` and `81^(sin^(2) theta ) + 81^(cos^(2) theta ) = 30` then `theta ` is

Options

A. `30^(@)`
B. `60^(@)`
C. `120^(@)`
D. `150^(@)`

Answer

To solve the equation \( 81^{\sin^2 \theta} + 81^{\cos^2 \theta} = 30 \) for \( \theta \) in the range \( 0 \leq \theta \leq \pi \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 81^{\sin^2 \theta} + 81^{\cos^2 \theta} = 30 \]
Since \( 81 = 3^4 \), we can rewrite the equation as:
\[ (3^4)^{\sin^2 \theta} + (3^4)^{\cos^2 \theta} = 30 \]
This simplifies to:
\[ 3^{4\sin^2 \theta} + 3^{4\cos^2 \theta} = 30 \]

### Step 2: Substitute variables
Let \( x = 3^{4\sin^2 \theta} \). Then, we can express \( 3^{4\cos^2 \theta} \) as:
\[ 3^{4\cos^2 \theta} = 3^{4(1 - \sin^2 \theta)} = \frac{3^4}{x} = \frac{81}{x} \]
Now, substitute this back into the equation:
\[ x + \frac{81}{x} = 30 \]

### Step 3: Multiply through by \( x \)
To eliminate the fraction, multiply through by \( x \):
\[ x^2 + 81 = 30x \]
Rearranging gives us a quadratic equation:
\[ x^2 - 30x + 81 = 0 \]

### Step 4: Solve the quadratic equation
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -30, c = 81 \):
\[ x = \frac{30 \pm \sqrt{(-30)^2 - 4 \cdot 1 \cdot 81}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{30 \pm \sqrt{900 - 324}}{2} = \frac{30 \pm \sqrt{576}}{2} = \frac{30 \pm 24}{2} \]
This gives us two solutions:
\[ x = \frac{54}{2} = 27 \quad \text{and} \quad x = \frac{6}{2} = 3 \]

### Step 5: Back substitute for \( \sin^2 \theta \)
Now we have two cases for \( x \):
1. \( x = 27 \)
2. \( x = 3 \)

#### Case 1: \( x = 27 \)
\[ 3^{4\sin^2 \theta} = 27 \implies 3^{4\sin^2 \theta} = 3^3 \implies 4\sin^2 \theta = 3 \implies \sin^2 \theta = \frac{3}{4} \]
Thus, \( \sin \theta = \frac{\sqrt{3}}{2} \), which gives:
\[ \theta = 60^\circ \quad \text{or} \quad \theta = 120^\circ \]

#### Case 2: \( x = 3 \)
\[ 3^{4\sin^2 \theta} = 3 \implies 4\sin^2 \theta = 1 \implies \sin^2 \theta = \frac{1}{4} \]
Thus, \( \sin \theta = \frac{1}{2} \), which gives:
\[ \theta = 30^\circ \quad \text{or} \quad \theta = 150^\circ \]

### Final Solution
The possible values for \( \theta \) are:
\[ \theta = 30^\circ, 60^\circ, 120^\circ, 150^\circ \]