Question

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11

hence solve the equation \\(\frac{\sin \theta + 2 \cos \theta}{\cos \theta - 2 \sin \theta} - \frac{\sin \theta - 2 \cos \theta}{\cos \theta + 2 \sin \theta} = 5\\) for \\(0^\circ < \theta < 180^\circ\\).

Explanation:

Simplify the left-hand side expression

Using the Algebraic Simplification of Trigonometric Fractions knowledge point

$$
LATEXBLOCK0
$$

Expand the numerator and denominator

Using the Trigonometric Identities knowledge point

$$
LATEXBLOCK1
$$

Set up the simplified equation

Using the Trigonometric Identities knowledge point

$$
LATEXBLOCK2
$$

Solve for sine of theta

Using the Trigonometric Identities knowledge point

$$
LATEXBLOCK3
$$

Determine the valid angles in the given interval

To solve for \(\theta\) in the interval \(0^\circ < \theta < 180^\circ\), we note that \(\sin\theta\) must be positive.
Therefore, we only consider:

$$ \sin\theta = \frac{1}{5} = 0.2 $$

Calculating the principal value:

$$ \theta_1 = \arcsin(0.2) \approx 11.5^\circ $$

Finding the second solution in the second quadrant:

$$ \theta_2 = 180^\circ - 11.536^\circ \approx 168.5^\circ $$

Both values satisfy the original equation's domain restrictions where denominators are non-zero.

Answer:

$$ \theta \approx 11.5^\circ \quad \text{or} \quad \theta \approx 168.5^\circ $$