Question

find the values of \\( \sin t, \cos t, \tan t, \csc t, \sec t \\), and \\( \cot t \\) if \\( p=\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\
ight) \\) is the point on the unit circle that corresponds to the real number \\( t \\).
\\( \sin t= \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression )
\\( \cos t= \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression )
\\( \tan t= \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression )

Explanation:

Step1: Recall the formula for tangent

The formula for \(\tan t=\frac{\sin t}{\cos t}\).

Step2: Substitute the values of \(\sin t\) and \(\cos t\)

Given \(\sin t =-\frac{1}{2}\) and \(\cos t=-\frac{\sqrt{3}}{2}\), then \(\tan t=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}\).

Step3: Simplify the expression

When dividing \(\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}\), the \(- 2\) in the numerator and denominator cancels out, and we get \(\tan t=\frac{1}{\sqrt{3}}\). Rationalizing the denominator (multiply numerator and denominator by \(\sqrt{3}\)), we have \(\tan t=\frac{\sqrt{3}}{3}\).

Answer:

\(\tan t=\frac{\sqrt{3}}{3}\)