The sum of 330^(∘) and 30^(∘) equals a full circle.
B
b
The sum of 120^(∘) and 60^(∘) equals a semicircle.
C
c
The sum of 113^(∘) and 67^(∘) equals a semicircle.
D
d
The sum of 180^(∘) and 23^(∘) equals 203.
A
a
30^(∘)
B
b
60^(∘)
C
c
67^(∘)
D
d
23^(∘)
Practice makes perfect
a
A rotation of 330^(∘) puts us in the fourth quadrant. Let's illustrate this.
Notice that a rotation of 330^(∘) around the unit circle is the same thing as a rotation of - 30^(∘) starting from 0^(∘).
Therefore, we can use a reference angle of 30^(∘) in the first quadrant instead.
b
An angle of 120^(∘) puts us in the second quadrant.
A rotation of 120^(∘) is 60^(∘) shy of 180^(∘). This means a rotation of 60^(∘) clockwise, starting from 180^(∘), is the same thing as a rotation of 120^(∘) counterclockwise when starting from 0^(∘). Therefore, the reference angle we should use in the first quadrant is 60^(∘).
c
An angle of 113^(∘) puts us in the second quadrant.
A rotation of 113^(∘) is 67^(∘) shy of 180^(∘). This means a rotation of 67^(∘) clockwise, starting from 180^(∘), is the same thing as a rotation of 113^(∘) counterclockwise starting from 0^(∘). Therefore, the reference angle we should use in the first quadrant is 67^(∘).
d
An angle of 203^(∘) puts us in the third quadrant.
A rotation of 203^(∘) can be viewed as a rotation of 180^(∘) followed by a rotation of 23^(∘)
If we extend the black segment in the opposite direction, we create an angle of 23^(∘) in the first quadrant. To prove that 23^(∘) is the reference angle we should use, we will highlight the triangles that the segments create with the unit circle and the horizontal axis.
Since both of the triangles have two pairs of congruent angles and their hypotenuses are both 1, they are congruent according to the AAS Congruence Theorem. Therefore, they must have two pairs of congruent legs, which means sin 23^(∘) and cos 23^(∘) can be used to find the sine and cosine value of 203^(∘).
