This diagram tells us that when an angle θ is in standard position with its terminal sideintersecting the unit circle at (x,y), then x=cosθ and y=sinθ. Now, consider the given diagram.
We are asked to determine the sign of sinθ, cosθ, and tanθ. We notice that the y-value of the point of intersection is positive and its x-value is negative. This means that sinθ is positive and that cosθ is negative.
Trigonometric Function
Sign
sinθ
+
cosθ
-
Next, we recall that the tangent function is the quotient of the sine and cosine functions.
tanθ=sinθ/cosθ
The quotient of a positive number and a negative number is negative. This means that tanθ is negative in this case. Let's include this information in our table!
Trigonometric Function
Sign
sinθ
+
cosθ
-
tanθ
-
b We are now asked to find in which quadrant lies the terminal side of -θ. We will first write down the quadrants in their respective position.
In order to draw the negative of the given angle, we start by placing the initial side in the same place. To place the terminal side we move clockwise instead of moving counter-clockwise.
We notice that the terminal side of -θ lies in Quadrant III.
c Let's take a look at the angle -θ that we drew in Part B.
This time we notice that the y-value and the x-values of the point of intersection are both negative. This means that sinθ and cosθ are both negative. Since the quotient of two negative numbers is positive, then tanθ is positive in this case.
