Question

solve the equation for all exact solutions where appropriate
round approximate answers in degrees to the nearest tenth
write answers using the least possible nonnegative angle
measures

\\( \sin \theta \sec \theta - \sin \theta = 0 \\)

choose the correct answer below

\\( \bigcirc \\) a. \\( \\{ 0 ^ { \circ } \\} \\)
\\( \bigcirc \\) b. \\( \\{ 90 ^ { \circ } n \\), where \\( n \\) is any integer \\}
\\( \bigcirc \\) c. \\( \\{ 270 ^ { \circ } n \\), where \\( n \\) is any integer \\}
\\( \bigcirc \\) d. \\( \\{ 180 ^ { \circ } n \\), where \\( n \\) is any integer \\}

Explanation:

Step1: Simplify the equation

Use the identity \(\sec\theta=\frac{1}{\cos\theta}\).
The given equation \(\sin\theta\sec\theta - \sin\theta = 0\) becomes \(\sin\theta\frac{1}{\cos\theta}-\sin\theta = 0\).
Factor out \(\sin\theta\): \(\sin\theta(\frac{1}{\cos\theta}- 1)=0\).

Step2: Set each factor equal to zero

Case 1: \(\sin\theta = 0\).
The general solution for \(\sin\theta=0\) is \(\theta = 180^{\circ}n\), where \(n\in\mathbb{Z}\) (using the unit - circle, \(\sin\theta = y\) - coordinate on the unit circle, and \(y = 0\) when \(\theta=0^{\circ},180^{\circ},360^{\circ},\cdots\)).
Case 2: \(\frac{1}{\cos\theta}-1 = 0\).
\(\frac{1}{\cos\theta}=1\), then \(\cos\theta = 1\). The general solution for \(\cos\theta = 1\) is also \(\theta = 360^{\circ}n=180^{\circ}(2n)\) (\(n\in\mathbb{Z}\)).

Answer:

D. \(\{180^{\circ}n,\text{ where }n\text{ is any integer}\}\)