Questions: Solve the equation. 9+9 sin theta=6 cos^2 theta

Solution Steps

To solve the equation \(9 + 9 \sin \theta = 6 \cos^2 \theta\), we can use trigonometric identities to simplify and solve for \(\theta\). First, use the identity \(\cos^2 \theta = 1 - \sin^2 \theta\) to express everything in terms of \(\sin \theta\). This will transform the equation into a quadratic form in terms of \(\sin \theta\). Then, solve the quadratic equation to find the possible values of \(\sin \theta\), and finally determine the corresponding angles \(\theta\).

The original equation is: \[ 9 + 9 \sin \theta = 6 \cos^2 \theta \] Using the identity \(\cos^2 \theta = 1 - \sin^2 \theta\), we can rewrite the equation as: \[ 9 + 9 \sin \theta = 6(1 - \sin^2 \theta) \] Simplifying, we get: \[ 9 + 9 \sin \theta = 6 - 6 \sin^2 \theta \]

Rearrange the equation to form a quadratic equation in terms of \(\sin \theta\): \[ 6 \sin^2 \theta + 9 \sin \theta + 3 = 0 \]

The quadratic equation \(6 \sin^2 \theta + 9 \sin \theta + 3 = 0\) can be solved using the quadratic formula: \[ \sin \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 6\), \(b = 9\), and \(c = 3\). Solving this gives: \[ \sin \theta = -1 \quad \text{and} \quad \sin \theta = -\frac{1}{2} \]

For \(\sin \theta = -1\), the angle is: \[ \theta = -\frac{\pi}{2} \] For \(\sin \theta = -\frac{1}{2}\), the angle is: \[ \theta = -\frac{\pi}{6} \]

Final Answer