Trigonometric Equations

Factoring the equation, we get ((2\sin\theta - 1)(\sin\theta + 1) = 0). Solving each factor separately, we find (\sin\theta = \frac{1}{2}) or (\sin\theta = -1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{3}, \frac{5\pi}{3}).

Factoring the equation, we get ((\tan\theta - 2)(\tan\theta + 1) = 0). Solving each factor separately, we find (\tan\theta = 2) or (\tan\theta = -1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{4}, \frac{3\pi}{4}).

Using the identity (\cos^2\theta = 1 - \sin^2\theta), we can rewrite the equation as (2(1 - \sin^2\theta) + 3\sin\theta - 5 = 0). Expanding and rearranging, we get (-2\sin^2\theta + 3\sin\theta - 3 = 0). Factoring, we find ((2\sin\theta - 3)(\sin\theta - 1) = 0). Solving each factor separately, we find (\sin\theta = \frac{3}{2}) or (\sin\theta = 1). Since (\sin\theta) cannot be greater than 1, we discard the first solution. Using the unit circle or reference angles, we find the solution (\theta = \frac{\pi}{6}, \frac{5\pi}{6}).

Using the identity (\sec^2\theta = 1 + \tan^2\theta), we can rewrite the equation as (1 + \tan^2\theta - 2\tan\theta - 3 = 0). Expanding and rearranging, we get (\tan^2\theta - 2\tan\theta - 4 = 0). Factoring, we find ((\tan\theta - 4)(\tan\theta + 1) = 0). Solving each factor separately, we find (\tan\theta = 4) or (\tan\theta = -1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{3}, \frac{5\pi}{3}).

Using the identity (\sin^2\theta + \cos^2\theta = 1), we can simplify the equation to (1 - 2\sin\theta\cos\theta = 1). Rearranging, we get (\sin\theta\cos\theta = 0). This means either (\sin\theta = 0) or (\cos\theta = 0). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{4}, \frac{3\pi}{4}).

Factoring the equation, we get ((2\sin\theta - 1)(\sin\theta - 1) = 0). Solving each factor separately, we find (\sin\theta = \frac{1}{2}) or (\sin\theta = 1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{6}, \frac{5\pi}{6}).

Factoring the equation, we get ((\tan\theta + 1)^2 = 0). Solving for (\tan\theta), we find (\tan\theta = -1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{4}, \frac{3\pi}{4}).

Using the identity (\sec^2\theta = 1 + \tan^2\theta), we can simplify the equation to (1 + \tan^2\theta - \tan^2\theta = 1). This simplifies to (1 = 1), which is true for all values of (\theta). Therefore, the equation has infinitely many solutions in the interval ([0, 2\pi)).

Using the identity (\sin^2\theta + \cos^2\theta = 1), we can rewrite the equation as (\sin^2\theta - (1 - \sin^2\theta) = \frac{1}{2}). Simplifying, we get (2\sin^2\theta - 1 = \frac{1}{2}). Solving for (\sin\theta), we find (\sin\theta = \pm\frac{\sqrt{6}}{4}). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{4}, \frac{3\pi}{4}).

Factoring the equation, we get ((\cot\theta - 2)(\cot\theta - 1) = 0). Solving each factor separately, we find (\cot\theta = 2) or (\cot\theta = 1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{6}, \frac{5\pi}{6}).

Using the identity (\cos^2\theta = 1 - \sin^2\theta), we can rewrite the equation as (2(1 - \sin^2\theta) + \sin\theta - 1 = 0). Expanding and rearranging, we get (-2\sin^2\theta + \sin\theta - 1 = 0). Factoring, we find ((2\sin\theta - 1)(\sin\theta - 1) = 0). Solving each factor separately, we find (\sin\theta = \frac{1}{2}) or (\sin\theta = 1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{6}, \frac{5\pi}{6}).

Using the identity (\sec^2\theta = 1 + \tan^2\theta), we can rewrite the equation as (1 + \tan^2\theta + \tan\theta - 1 = 0). Simplifying, we get (\tan^2\theta + \tan\theta = 0). Factoring, we find (\tan\theta(\tan\theta + 1) = 0). Solving each factor separately, we find (\tan\theta = 0) or (\tan\theta = -1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{4}, \frac{3\pi}{4}).

Using the identity (\sin^2\theta + \cos^2\theta = 1), we can simplify the equation to (1 - \sin\theta - \cos\theta = 0). Rearranging, we get (\sin\theta + \cos\theta = 1). This suggests that (\sin\theta) and (\cos\theta) have the same sign. The only way this can happen is if both (\sin\theta) and (\cos\theta) are positive. Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{4}, \frac{3\pi}{4}).

Using the identity (\csc^2\theta = 1 + \cot^2\theta), we can rewrite the equation as (1 + \cot^2\theta - 2\csc\theta - 3 = 0). Expanding and rearranging, we get (\cot^2\theta - 2\csc\theta - 2 = 0). Factoring, we find ((\cot\theta - 2)(\cot\theta + 1) = 0). Solving each factor separately, we find (\cot\theta = 2) or (\cot\theta = -1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{6}, \frac{5\pi}{6}).

Factoring the equation, we get ((2\sin\theta - 1)(\sin\theta - 1) = 0). Solving each factor separately, we find (\sin\theta = \frac{1}{2}) or (\sin\theta = 1). Using the unit circle or reference angles, we find the solutions (\theta = \frac{\pi}{6}, \frac{5\pi}{6}).