Solve the following equations for `0^(@)lethetale90^(@)` :
(i) `2cos^(2)theta=(1)/(2)`
(ii) `4sin^(2)theta-3=0`
(iii) `sin^(2)theta-(1)/(2)sintheta=0`
(iv) `tan^(2)theta-(sqrt(3)+1)tantheta+sqrt(3)=0`
(v) `sintheta-costheta=0`

Answer

Let's solve the equations step by step.

### (i) Solve `2cos^(2)theta = (1)/(2)`

1. Start with the equation:
\[ 2\cos^2\theta = \frac{1}{2} \]

2. Divide both sides by 2:
\[ \cos^2\theta = \frac{1}{4} \]

3. Take the square root of both sides:
\[ \cos\theta = \pm \frac{1}{2} \]

4. Since we are looking for solutions in the range \(0^\circ \leq \theta \leq 90^\circ\), we only consider the positive value:
\[ \cos\theta = \frac{1}{2} \]

5. Find the angle:
\[ \theta = 60^\circ \]

### (ii) Solve `4sin^(2)theta - 3 = 0`

1. Start with the equation:
\[ 4\sin^2\theta - 3 = 0 \]

2. Rearrange to isolate \(\sin^2\theta\):
\[ 4\sin^2\theta = 3 \]

3. Divide both sides by 4:
\[ \sin^2\theta = \frac{3}{4} \]

4. Take the square root of both sides:
\[ \sin\theta = \pm \frac{\sqrt{3}}{2} \]

5. Again, in the range \(0^\circ \leq \theta \leq 90^\circ\), we only consider the positive value:
\[ \sin\theta = \frac{\sqrt{3}}{2} \]

6. Find the angle:
\[ \theta = 60^\circ \]

### (iii) Solve `sin^(2)theta - (1)/(2)sintheta = 0`

1. Start with the equation:
\[ \sin^2\theta - \frac{1}{2}\sin\theta = 0 \]

2. Factor out \(\sin\theta\):
\[ \sin\theta(\sin\theta - \frac{1}{2}) = 0 \]

3. Set each factor to zero:
- \(\sin\theta = 0\) gives \(\theta = 0^\circ\)
- \(\sin\theta - \frac{1}{2} = 0\) gives \(\sin\theta = \frac{1}{2}\)

4. Find the angle for \(\sin\theta = \frac{1}{2}\):
\[ \theta = 30^\circ \]

5. Thus, the solutions are:
\[ \theta = 0^\circ, 30^\circ \]

### (iv) Solve `tan^(2)theta - (sqrt(3)+1)tantheta + sqrt(3) = 0`

1. Let \(y = \tan\theta\), rewrite the equation:
\[ y^2 - (\sqrt{3} + 1)y + \sqrt{3} = 0 \]

2. Use the quadratic formula:
\[ y = \frac{-(\sqrt{3} + 1) \pm \sqrt{(\sqrt{3} + 1)^2 - 4\sqrt{3}}}{2} \]

3. Calculate the discriminant:
\[ (\sqrt{3} + 1)^2 - 4\sqrt{3} = 3 + 2\sqrt{3} + 1 - 4\sqrt{3} = 4 - 2\sqrt{3} \]

4. Solve for \(y\):
\[ y = \frac{\sqrt{3} + 1 \pm \sqrt{4 - 2\sqrt{3}}}{2} \]

5. Find the values of \(\theta\) corresponding to \(y = 1\) and \(y = \sqrt{3}\):
- \(\tan\theta = 1\) gives \(\theta = 45^\circ\)
- \(\tan\theta = \sqrt{3}\) gives \(\theta = 60^\circ\)

6. Thus, the solutions are:
\[ \theta = 45^\circ, 60^\circ \]

### (v) Solve `sintheta - costheta = 0`

1. Start with the equation:
\[ \sin\theta - \cos\theta = 0 \]

2. Rearrange to find:
\[ \sin\theta = \cos\theta \]

3. Divide both sides by \(\cos\theta\) (assuming \(\cos\theta \neq 0\)):
\[ \tan\theta = 1 \]

4. Find the angle:
\[ \theta = 45^\circ \]

### Summary of Solutions:
- (i) \( \theta = 60^\circ \)
- (ii) \( \theta = 60^\circ \)
- (iii) \( \theta = 0^\circ, 30^\circ \)
- (iv) \( \theta = 45^\circ, 60^\circ \)
- (v) \( \theta = 45^\circ \)