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

Explanation:

Step1: Simplify the equation

Given \(\sin\theta\sec\theta-\sin\theta = 0\). Since \(\sec\theta=\frac{1}{\cos\theta}\), the equation becomes \(\sin\theta\times\frac{1}{\cos\theta}-\sin\theta=0\), which is \(\frac{\sin\theta}{\cos\theta}-\sin\theta = 0\). Factor out \(\sin\theta\): \(\sin\theta(\frac{1}{\cos\theta}- 1)=0\). So either \(\sin\theta = 0\) or \(\frac{1}{\cos\theta}-1 = 0\).

• If \(\sin\theta = 0\), then \(\theta = 180^{\circ}n\), \(n\in\mathbb{Z}\) (because the sine function \(y = \sin x\) has zeros at \(x = k\pi\) or \(x=180^{\circ}k\), \(k\in\mathbb{Z}\)).
• If \(\frac{1}{\cos\theta}-1 = 0\), then \(\frac{1}{\cos\theta}=1\), so \(\cos\theta = 1\). When \(\cos\theta=1\), \(\theta = 360^{\circ}n=180^{\circ}\times2n\), \(n\in\mathbb{Z}\)

Answer:

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