Question

unit: trigonometry
progress:
the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer.
acute angle w has \\(\sin w = \frac{1}{2}\\) and \\(\sec w = \frac{2}{\sqrt{3}}\\). which are values of \\(\csc w\\) and \\(\cos w\\)?
\\(\csc w = 2\\) and \\(\cos w = \frac{1}{2}\\)
\\(\csc w = \sqrt{3}\\) and \\(\cos w = \frac{1}{2}\\)
\\(\csc w = \sqrt{3}\\) and \\(\cos w = \frac{\sqrt{3}}{2}\\)
\\(\csc w = 2\\) and \\(\cos w = \frac{\sqrt{3}}{2}\\)

Explanation:

Step1: Recall reciprocal identities

The reciprocal identity for sine and cosecant is \( \csc \theta=\frac{1}{\sin \theta} \), and for secant and cosine is \( \cos \theta = \frac{1}{\sec \theta} \).

Step2: Calculate \( \csc W \)

Given \( \sin W=\frac{1}{2} \), using \( \csc W=\frac{1}{\sin W} \), we substitute \( \sin W \):
\( \csc W=\frac{1}{\frac{1}{2}} = 2 \).

Step3: Calculate \( \cos W \)

Given \( \sec W=\frac{2}{\sqrt{3}} \), using \( \cos W=\frac{1}{\sec W} \), we substitute \( \sec W \):
\( \cos W=\frac{1}{\frac{2}{\sqrt{3}}}=\frac{\sqrt{3}}{2} \) (rationalizing the denominator is not needed here as we can simplify directly).

Answer:

\( \csc W = 2 \) and \( \cos W = \frac{\sqrt{3}}{2} \) (the last option, e.g., if options are labeled as D: \( \csc W = 2 \) and \( \cos W = \frac{\sqrt{3}}{2} \), then the answer is D. \( \csc W = 2 \) and \( \cos W = \frac{\sqrt{3}}{2} \))