Question

Line: y= -x Quadrant: 2Trig: The terminal side of (theta) lies on the given line in the specified quadrant. Find the exact values of the six trigonometric functions of (theta) by finding a point on the line.Sin(theta)=Cos(theta)=Tan(theta)=Csc(theta)=Sec(theta)=Cot(theta)=

Answer 1

The exact values of the six trigonometric functions are;

`\begin{gathered} \sin \theta=(1)/(√(2)) \\ \cos \theta=(-1)/(√(2)) \\ \tan \theta=-1 \\ \text{csc }\theta=√(2) \\ \text{sec }\theta=-√(2) \\ \text{cot }\theta=-1 \end{gathered}`

Step-by-step explanation:

Given the function;

`y=-x`

At y = 1 the value of x is;

`\begin{gathered} y=-x \\ 1=-x \\ x=-1 \end{gathered}`

Therefore, at x=-1, y=1. (-1,1)

So, sketching the coordinates we have;

From the sketch, we can see that;

`\begin{gathered} Opposite=y=1 \\ \text{Adjacent = x = -1} \\ \text{Hypotenuse = }√((-1)^2+1^2)=√(2) \end{gathered}`

We can used the following values to find the values of each of the following;

`\begin{gathered} \sin \theta=(opposite)/(hypotenuse)=(1)/(√(2)) \\ \cos \theta=\frac{Adjacent}{\text{hypotenuse}}=(-1)/(√(2)) \\ \tan \theta=(opposite)/(adjacent)=(1)/(-1)=-1 \\ \text{csc }\theta=(1)/(\sin \theta)=(√(2))/(1)=√(2) \\ \text{sec }\theta=(1)/(\cos \theta)=(√(2))/(-1)=-√(2) \\ \text{cot }\theta=(1)/(\tan \theta)=(-1)/(1)=-1 \end{gathered}`

Therefore, the exact values of the six trigonometric functions are;

`\begin{gathered} \sin \theta=(1)/(√(2)) \\ \cos \theta=(-1)/(√(2)) \\ \tan \theta=-1 \\ \text{csc }\theta=√(2) \\ \text{sec }\theta=-√(2) \\ \text{cot }\theta=-1 \end{gathered}`