Question

using your knowledge about the unit circle, calculate the cosecant, secant, and cotangent values of the angle below.
find \\(\csc(t)\\), \\(\sec(t)\\) and \\(\cot(t)\\) when \\(t = \pi\\) radians.
show your work here
hint to add the square root symbol (\\(\sqrt{\square}\\)), type
oot\
\\(\csc(t) =\\)
\\(\sec(t) =\\)
\\(\cot(t) =\\)

Explanation:

Step1: Recall unit circle coordinates for \( t = \pi \)

On the unit circle, for \( t=\pi \) radians (180 degrees), the coordinates are \( (\cos(t), \sin(t)) = (-1, 0) \). So, \( \sin(\pi) = 0 \) and \( \cos(\pi)= -1 \).

Step2: Calculate \( \csc(t) \) (cosecant)

The formula for cosecant is \( \csc(t)=\frac{1}{\sin(t)} \). Substituting \( \sin(\pi) = 0 \), we get \( \csc(\pi)=\frac{1}{0} \), which is undefined (since division by zero is not allowed).

Step3: Calculate \( \sec(t) \) (secant)

The formula for secant is \( \sec(t)=\frac{1}{\cos(t)} \). Substituting \( \cos(\pi) = -1 \), we have \( \sec(\pi)=\frac{1}{-1}=-1 \).

Step4: Calculate \( \cot(t) \) (cotangent)

The formula for cotangent is \( \cot(t)=\frac{\cos(t)}{\sin(t)} \). Substituting \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \), we get \( \cot(\pi)=\frac{-1}{0} \), which is undefined (division by zero).

Answer:

\( \csc(t) \): undefined
\( \sec(t) = -1 \)
\( \cot(t) \): undefined