To solve the equation:

$\csc^2(\theta) = 7\cot(\theta) + 9$

Step 1: Use trigonometric identities

We know the identity: $\csc^2(\theta) = 1 + \cot^2(\theta)$ Substitute this into the equation: $1 + \cot^2(\theta) = 7\cot(\theta) + 9$

Step 2: Rearrange into a quadratic equation

$\cot^2(\theta) - 7\cot(\theta) - 8 = 0$

Let $x = \cot(\theta)$, then the equation becomes: $x^2 - 7x - 8 = 0$

Step 3: Solve the quadratic equation

Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = -7$, and $c = -8$. Substitute these values: $x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-8)}}{2(1)}$ $x = \frac{7 \pm \sqrt{49 + 32}}{2}$ $x = \frac{7 \pm \sqrt{81}}{2}$ $x = \frac{7 \pm 9}{2}$

So, the two solutions are: $x = \frac{7 + 9}{2} = 8, \quad x = \frac{7 - 9}{2} = -1$

Step 4: Back-substitute $x = \cot(\theta)$

• When $\cot(\theta) = 8$: $\theta = \cot^{-1}(8)$ Using a calculator, this gives: $\theta = 0.124 \, \text{radians} + k\pi, \quad k \in \mathbb{Z}$

• When $\cot(\theta) = -1$: $\theta = \cot^{-1}(-1)$ Using a calculator, this gives: $\theta = 2.356 \, \text{radians} + k\pi, \quad k \in \mathbb{Z}$

Final Answer:

$\theta = 0.124 + k\pi, \, 2.356 + k\pi \quad \text{where } k \in \mathbb{Z}.$

Would you like further details or have any questions?

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Related Questions:

1. What is the general solution for $\cot(\theta) = c$?
2. How does the identity (\