Cogito
AP Precalculus · Chapter 6 · Lesson 1
The Unit Circle and Radians
Freeing the ratios from the triangle.
12 problems · about 21 minutes · AP Precalculus 3.3, 3.4
What this lesson teaches
The student defines sine and cosine on the unit circle and converts between degrees and radians.
- On the unit circle the terminal point is (cos θ, sin θ).
- Every angle has a terminal point, so obtuse and negative angles are included.
- A full turn is 2π radians, and radians make the formulas clean.
Warm Up
Straightforward practice. Get the method working first.
5 problemsWhat is sin 180°?
Answer 0
Why 0.
Why does calculus use radians rather than degrees?
Answer Arc length and the calculus formulas come out clean, with no conversion factor.
Why The formulas need no extra factor.
What is sin 90°?
Answer 1
Why The point is (0, 1).
What is cos 180°?
Answer -1
Why The point is (−1, 0).
What is sin 0°?
Answer 0
Why The y-coordinate.
Build It Up
The same ideas with more to keep track of.
3 problemsHow many degrees is 2π radians?
Answer 360
Why A full turn.
How many degrees is π ÷ 2 radians?
Answer 90
Why A quarter turn.
Radius 5, angle 3 radians. Arc length?
Answer 15
Why rθ.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each quantity by which coordinate it reads.
Answer The x-coordinate: Cosine, Horizontal position on the circle · The y-coordinate: Sine, Vertical position on the circle
Why Cosine goes across.
What is cos 90°?
Answer 0
Why The x-coordinate at the top.
The Conversion: How many degrees is π radians?
Answer 180 degrees
Why 180°.
The Coordinate: On the unit circle, what is cos 0°?
Answer 1
Why 1.