Question

find the exact value of the following expression without using a calculator.

csc \frac{\pi}{6}

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

\\( \bigcirc \mathrm { a } \\).

\\( csc \frac { pi } { 6 } = \\)

(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

\\( \bigcirc \mathrm { b } \\). the function is undefined.

Explanation:

Step1: Recall the reciprocal identity

The cosecant function is the reciprocal of the sine function, i.e., \(\csc\theta=\frac{1}{\sin\theta}\). So, \(\csc\frac{\pi}{6}=\frac{1}{\sin\frac{\pi}{6}}\).

Step2: Find the value of \(\sin\frac{\pi}{6}\)

We know that for the unit - circle or the 30 - 60 - 90 triangle (where \(\theta = 30^{\circ}=\frac{\pi}{6}\) radians), \(\sin\frac{\pi}{6}=\frac{1}{2}\).

Step3: Calculate \(\csc\frac{\pi}{6}\)

Substitute \(\sin\frac{\pi}{6}=\frac{1}{2}\) into the formula \(\csc\frac{\pi}{6}=\frac{1}{\sin\frac{\pi}{6}}\). Then \(\csc\frac{\pi}{6}=\frac{1}{\frac{1}{2}}\). Using the rule for dividing by a fraction (\(\frac{a}{\frac{b}{c}}=a\times\frac{c}{b}\), here \(a = 1\), \(b = 1\), \(c = 2\)), we get \(\csc\frac{\pi}{6}=2\).

Answer:

A. \(\csc\frac{\pi}{6}=2\)