Question

find the values of all six trig function of $2\pi$.

Explanation:

Step1: Recall the unit - circle definitions

On the unit circle \(x = r\cos\theta\), \(y = r\sin\theta\), and \(r = 1\). For \(\theta=2\pi\), the point on the unit circle is \((1,0)\).

Step2: Calculate sine and cosine

By the definition \(\sin\theta=\frac{y}{r}\), \(\cos\theta=\frac{x}{r}\). Since \(x = 1\), \(y = 0\), \(r = 1\), we have \(\sin(2\pi)=\frac{0}{1}=0\), \(\cos(2\pi)=\frac{1}{1}=1\).

Step3: Calculate tangent

Using the formula \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), substituting \(\sin(2\pi) = 0\) and \(\cos(2\pi)=1\), we get \(\tan(2\pi)=\frac{0}{1}=0\).

Step4: Calculate cosecant

Using the formula \(\csc\theta=\frac{1}{\sin\theta}\), since \(\sin(2\pi) = 0\), \(\csc(2\pi)\) is undefined (because division by zero is not allowed).

Step5: Calculate secant

Using the formula \(\sec\theta=\frac{1}{\cos\theta}\), substituting \(\cos(2\pi)=1\), we get \(\sec(2\pi)=\frac{1}{1}=1\).

Step6: Calculate cotangent

Using the formula \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), since \(\sin(2\pi) = 0\), \(\cot(2\pi)\) is undefined (because division by zero is not allowed).

Answer:

\(\sin(2\pi)=0\), \(\cos(2\pi)=1\), \(\tan(2\pi)=0\), \(\csc(2\pi)\) is undefined, \(\sec(2\pi)=1\), \(\cot(2\pi)\) is undefined.