Question

fatima wants to find the value of \\( \sin \theta \\), given \\( \cot \theta = \frac{4}{7} \\). which identity would be best for fatima to use?
\\( \circ \\) \\( \cos \theta = \frac{1}{\sec \theta} \\)
\\( \circ \\) \\( \sin^2 \theta + \cos^2 \theta = 1 \\)
\\( \circ \\) \\( \csc \theta = \frac{r}{y} \\)
\\( \circ \\) \\( 1 + \cot^2 \theta = \csc^2 \theta \\)

Explanation:

Step1: Analyze given info

We know \(\cot\theta=\frac{4}{7}\), need to find \(\sin\theta\). Recall trigonometric identities.

Step2: Evaluate each option

• Option 1: \(\cos\theta = \frac{1}{\sec\theta}\) relates \(\cos\) and \(\sec\), not helpful for \(\sin\) with \(\cot\).
• Option 2: \(\sin^{2}\theta+\cos^{2}\theta = 1\) needs \(\cos\) or \(\sin\), but we have \(\cot\) (which is \(\frac{\cos}{\sin}\)), not direct.
• Option 3: \(\csc\theta=\frac{r}{y}\) is a coordinate - based definition, not useful with \(\cot\) value.
• Option 4: The identity \(1 + \cot^{2}\theta=\csc^{2}\theta\) connects \(\cot\) (which we know) to \(\csc\) (and \(\csc=\frac{1}{\sin}\)). Since we know \(\cot\theta=\frac{4}{7}\), we can use \(1+\cot^{2}\theta=\csc^{2}\theta\) to find \(\csc\theta\), then find \(\sin\theta=\frac{1}{\csc\theta}\).

Answer:

\(1 + \cot^{2}\theta=\csc^{2}\theta\) (the fourth option: \(1+\cot^{2}\theta=\csc^{2}\theta\))