If `thetagt0` be an acute angle , then the value of `theta` in degree stisfying `(cos^(2)theta-3costheta+2)/(sin^(2)theta)=1` is

Options

A. `90^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Answer

To solve the equation

\[ \frac{\cos^2 \theta - 3 \cos \theta + 2}{\sin^2 \theta} = 1 \]

we will follow these steps:

### Step 1: Rewrite the equation

We can start by rewriting the equation:

\[ \cos^2 \theta - 3 \cos \theta + 2 = \sin^2 \theta \]

### Step 2: Use the Pythagorean identity

Using the identity \(\sin^2 \theta = 1 - \cos^2 \theta\), we can substitute for \(\sin^2 \theta\):

\[ \cos^2 \theta - 3 \cos \theta + 2 = 1 - \cos^2 \theta \]

### Step 3: Rearrange the equation

Now, we can rearrange the equation:

\[ \cos^2 \theta + \cos^2 \theta - 3 \cos \theta + 2 - 1 = 0 \]

This simplifies to:

\[ 2\cos^2 \theta - 3 \cos \theta + 1 = 0 \]

### Step 4: Solve the quadratic equation

Now we can solve the quadratic equation \(2x^2 - 3x + 1 = 0\) where \(x = \cos \theta\). We can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting \(a = 2\), \(b = -3\), and \(c = 1\):

\[ x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} \]

Calculating the discriminant:

\[ x = \frac{3 \pm \sqrt{9 - 8}}{4} \]

\[ x = \frac{3 \pm 1}{4} \]

This gives us two possible values:

\[ x_1 = \frac{4}{4} = 1 \quad \text{and} \quad x_2 = \frac{2}{4} = \frac{1}{2} \]

### Step 5: Find the angles

Now we find \(\theta\) corresponding to these values of \(x\):

1. For \(x_1 = 1\):
\[ \cos \theta = 1 \implies \theta = 0^\circ \]
(not an acute angle)

2. For \(x_2 = \frac{1}{2}\):
\[ \cos \theta = \frac{1}{2} \implies \theta = 60^\circ \]

### Conclusion

Thus, the value of \(\theta\) in degrees satisfying the equation is:

\[ \theta = 60^\circ \]

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