An angle of - 90^(∘) means we have to rotate clockwise on the unit circle.
D
d
Notice that 510^(∘) = 360^(∘) + 150^(∘)
E
e
The cosine value is given by the horizontal axis.
F
f
The tangent value is the ratio of sine to cosine.
A
a
sin 180^(∘) = 0
B
b
sin 360^(∘) = 0
C
c
sin - 90^(∘) = - 1
D
d
sin 510^(∘) = 1/2
E
e
cos 90^(∘) = 0
F
f
Undefined.
Practice makes perfect
a
In the unit circle, the sine value is given by the vertical axis. y=sin θ
To evaluate the given trig expression, we have to figure out which y-value corresponds to a rotation of 180^(∘). Let's illustrate this angle on the unit circle.
As we can see, sin 180^(∘) =0.
b
Let's draw an angle of 360^(∘) and identify what sine value it corresponds to.
As we can see, sin 360^(∘) =0.
c
For positive angles, the rotation around the circle is drawn counterclockwise. Therefore, an angle of - 90^(∘) must be drawn clockwise.
As we can see, sin - 90^(∘) = - 1.
d
To figure out the value of sin(510^(∘)), we recognize that 510^(∘) equals the sum of 360^(∘) and 150^(∘). Additionally, the reference angle to 150^(∘) is 30^(∘), which has a sine value of 12.
As we can see, sin 510^(∘) = 12.
e
In the unit circle, the cosine value is given by the horizontal axis.
x=cos θ
Therefore, to evaluate the given trig expression we have to figure out which x-value corresponds to a rotation of 90^(∘). Let's illustrate this angle on the unit circle.
As we can see cos 90^(∘) = 0.
f
Notice that the tangent of an angle is defined as the ratio of the angles sine to cosine.
tan θ = sin θ/cos θFrom Part C, we know that sin- 90^(∘) =0. We also have to determine cos - 90^(∘).
Now that we know the value of cos - 90^(∘), we can attempt to calculate tan - 90.
