How many values of \theta will satisfy the equation (\sin^{2}\theta-4\sin \theta+3)(4-\cos^{2}\theta+4\sin \theta)=0, where 0 \lt \theta \lt \frac{\pi}{2}?

Explanation

Set each factor to zero. First factor: \sin^2\theta - 4\sin\theta + 3 = 0 \implies (\sin\theta-1)(\sin\theta-3)=0. Thus \sin\theta=1 or 3. Since \theta \in (0, \pi/2), \sin\theta \lt 1, yielding no solution. Second factor: 4 - (1-\sin^2\theta) + 4\sin\theta = 0 \implies \sin^2\theta + 4\sin\theta + 3 = 0 \implies (\sin\theta+1)(\sin\theta+3)=0. Thus \sin\theta=-1 or -3, neither of which is possible in the interval. Hence, there are no solutions.

Related questions on Trigonometry

• Two poles are situated 24 m apart and their heights differ by 10 m. What is the distance between their tips?
• If \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 4, then which one of the following is a value of $(\tan^2 \...
• For 0 < \theta < \frac{\pi}{2}, consider the following : I. $(\tan^4 \theta + \tan^6 \theta)(\cot^4 \theta + \cot^6 \theta) = \sec^2 \the...
• If 3\sin \theta + 4\cos \theta = 5, then what is a value of 4\tan \theta + 3\cot \theta ?
• At a point on level ground, the tangent of the angle of elevation of the top of a tower is found to be \frac{5}{6}. On walking 70 m toward...