Question

evaluate the following trigonometric expressions. if an answer is undefined, enter dne.

\\( \sin ( 0 ) = \\)

\\( \cos ( 0 ) = \\)

\\( \tan ( 0 ) = \\)

\\( \cot ( 0 ) = \\)

\\( \sec ( 0 ) = \\)

\\( \csc ( 0 ) = \\)

question help: video message instructor post to forum

Explanation:

Step1: Recall the unit - circle definitions

On the unit circle \(x = \cos\theta\), \(y=\sin\theta\), and \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), \(\cot\theta = \frac{\cos\theta}{\sin\theta}\), \(\sec\theta=\frac{1}{\cos\theta}\), \(\csc\theta=\frac{1}{\sin\theta}\)

When \(\theta = 0\), on the unit - circle, the point is \((x,y)=(1,0)\)

Step2: Calculate \(\sin(0)\)

Since \(y = \sin\theta\), when \(\theta=0\), \(\sin(0)=0\)

Step3: Calculate \(\cos(0)\)

Since \(x=\cos\theta\), when \(\theta = 0\), \(\cos(0)=1\)

Step4: Calculate \(\tan(0)\)

Since \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), substituting \(\sin(0) = 0\) and \(\cos(0)=1\), we get \(\tan(0)=\frac{0}{1}=0\)

Step5: Calculate \(\cot(0)\)

Since \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), substituting \(\sin(0) = 0\) and \(\cos(0)=1\), we have \(\cot(0)=\frac{1}{0}\), which is undefined.

Step6: Calculate \(\sec(0)\)

Since \(\sec\theta=\frac{1}{\cos\theta}\), substituting \(\cos(0)=1\), we get \(\sec(0)=\frac{1}{1}=1\)

Step7: Calculate \(\csc(0)\)

Since \(\csc\theta=\frac{1}{\sin\theta}\), substituting \(\sin(0) = 0\), we have \(\csc(0)=\frac{1}{0}\), which is undefined.

Answer:

\(\sin(0)=0\)
\(\cos(0)=1\)
\(\tan(0)=0\)
\(\cot(0)=\text{DNE}\)
\(\sec(0)=1\)
\(\csc(0)=\text{DNE}\)