http://www.ck12.org Chapter 5. Trigonometric Functions
logic and the pattern:^02 ,
√ 1
2 ,
√ 2
2 ,
√ 3
2 ,
√ 4
2.
Example C
Evaluate cos 60◦ using the unit circle and right triangle trigonometry. What is the connection between thex
coordinate of the point and the cosine of the angle?
Solution: The point on the unit circle for 60◦is
(
(^12) ,
√ 3
2)
and the point is one unit from the origin. This can be
represented as a 30-60-90 triangle.
Since cosine is adjacent over hypotenuse, cosine turns out to be exactly thexcoordinate^12.
Concept Problem Revisited
Thexvalue of a point along the unit circle corresponds to the cosine of the angle. Theyvalue of a point corresponds
to the sine of the angle. When the angles and points are memorized, simply recall thexorycoordinate.
When evaluating cos( 135 ◦)your thought process should be something like this:
You know 135◦goes with the point
(
−
√ 2
2 ,
√ 2
2)
and cosine is thexportion. So, cos( 135 ◦) =−√ 2
2.
When evaluating sin(−^53 π)your thought process should be something like this:
You know−^53 πgoes with the point
(
(^12) ,−√ 23
)
and sine is theyportion. So, sin(−^53 π)=−√ 3
2.
Vocabulary
Coterminal Anglesare sets of angles such as− 10 ◦, 350 ◦, 710 ◦that start at the positivex-axis and end at the same
terminal side. Since coterminal angles end at identical points along the unit circle, trigonometric expressions
involving coterminal angles are equivalent: sin− 10 ◦=sin 350◦=sin 710◦.