Answer:
The exact values of the six trigonometric functions are;
[tex]\begin{gathered} \sin \theta=\frac{1}{\sqrt{2}} \\ \cos \theta=\frac{-1}{\sqrt{2}} \\ \tan \theta=-1 \\ \text{csc }\theta=\sqrt{2} \\ \text{sec }\theta=-\sqrt{2} \\ \text{cot }\theta=-1 \end{gathered}[/tex]
Explanation:
Given the function;
[tex]y=-x[/tex]
At y = 1 the value of x is;
[tex]\begin{gathered} y=-x \\ 1=-x \\ x=-1 \end{gathered}[/tex]
Therefore, at x=-1, y=1. (-1,1)
So, sketching the coordinates we have;
From the sketch, we can see that;
[tex]\begin{gathered} Opposite=y=1 \\ \text{Adjacent = x = -1} \\ \text{Hypotenuse = }\sqrt{(-1)^2+1^2}=\sqrt{2} \end{gathered}[/tex]
We can used the following values to find the values of each of the following;
[tex]\begin{gathered} \sin \theta=\frac{opposite}{hypotenuse}=\frac{1}{\sqrt{2}} \\ \cos \theta=\frac{Adjacent}{\text{hypotenuse}}=\frac{-1}{\sqrt{2}} \\ \tan \theta=\frac{opposite}{adjacent}=\frac{1}{-1}=-1 \\ \text{csc }\theta=\frac{1}{\sin \theta}=\frac{\sqrt{2}}{1}=\sqrt{2} \\ \text{sec }\theta=\frac{1}{\cos \theta}=\frac{\sqrt{2}}{-1}=-\sqrt{2} \\ \text{cot }\theta=\frac{1}{\tan \theta}=\frac{-1}{1}=-1 \end{gathered}[/tex]
Therefore, the exact values of the six trigonometric functions are;
[tex]\begin{gathered} \sin \theta=\frac{1}{\sqrt{2}} \\ \cos \theta=\frac{-1}{\sqrt{2}} \\ \tan \theta=-1 \\ \text{csc }\theta=\sqrt{2} \\ \text{sec }\theta=-\sqrt{2} \\ \text{cot }\theta=-1 \end{gathered}[/tex]
