a Cosine is measured on the horizontal axis, and sine on the vertical axis.
B
b Draw y=- 1 in the unit circle.
C
c Cosine is measured on the horizontal axis, and sine on the vertical axis.
D
d Draw x=- 0.9 and y=0.4. Do they intersect on the unit circle's perimeter?
E
e Draw x= 0.9 and y=0.8. Do they intersect on the unit circle's perimeter?
A
aExample Solution:
B
bExample Solution:
C
cExample Solution:
D
dDiagram:
E
e No, it cannot. See solution.
Practice makes perfect
a In the unit circle, cosine is measured on the horizontal axis and sine on the vertical axis. If we want a positive cosine and a negative sine we have to draw an angle in the fourth quadrant, which means it has to fall in the interval 270^(∘)<θ<360^(∘). For example, 330^(∘) will have a positive cosine and a negative sine.
b In Part A, we explained that sine is measured on the vertical axis. Therefore, an angle that has a sine of -1 must be an angle that coincides with y=- 1.
The given angle covers three of four quadrants, which must mean that θ =270^(∘).
c Like in Part A, we know that cosine is measured on the horizontal axis, and sine on the vertical axis. If we want a negative cosine and a negative sine we have to sketch an angle in the third quadrant, which means it has to fall in the interval 180^(∘)<θ<270^(∘). For example, the angle 225^(∘) will have a negative cosine and a negative sine.
d We can mark a cosine of - 0.9 and a sine of about 0.4 by adding the vertical segment x=- 0.9 and the horizontal segment y=0.4 on the unit circle.
Where the two segments intersect is the angle that gives a cosine of -0.9 and a sine of about 0.4. Notice that the lines do not intersect on the unit circle's perimeter, which is why the angle only gives approximately x=- 0.9 and y=0.4.
e Like in Part D, we will add y=0.9 and x=0.8 to the unit circle.
As we can see, x=0.9 and y=0.8 do not intersect on the unit circle's perimeter. Therefore, there is no angle that gives a sine of 0.9 and a cosine of 0.8.
