The Unit Circle - Exercises | IB Math AAHL | Perplex

The Unit Circle

Sine, cosine & tangent functions in the unit circle

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Exercises

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Key Skills

Sine and Cosine on the Unit Circle

SL 3.5

The key idea with triangles who's hypotenuse lie on the unit circle (that form an angle of θ with the x-axis) is that cosθ represents length of the base, and sinθ represents the height.

Take a look at the graph below and notice the following relationships always hold:

cosθsinθ=x-coordinate=y-coordinate

Key values of Sin, Cos & Tan

SL 3.5

The following table shows the values of sinθ and cosθ for the so called critical angles θ. These are angles that give "nice" values for sin and cos:

θ (rad)	sinθ	cosθ
0	0	1
6π	21	2√3
4π	2√2	2√2
3π	2√3	21
2π	1	0

Quadrants

SL 3.5

The unit circle can be divided into quadrants based on the sign of cosθ and sinθ. These correspond to the 4 quadrants produced by the intersection of the x and y axes. The quadrants are denoted Q1, Q2, Q3 and Q4.

Quadrant	sin	cos
Q1	+	+
Q2	+	-
Q3	-	-
Q4	-	+

Periodicity

SL 3.5

Since a full circle is 2π radians, adding 2π to any angle θ gives the same point on the unit circle. In fact, adding any integer multiple of 2π gives the same point:

cos(θ+2kπ)sin(θ+2kπ)=cosθ=sinθ

Symmetry About the X-axis

SL 3.5

sin(−θ)cos(−θ)=−sinθ=cosθ

Symmetry About the Y-axis

SL 3.5

sin(π−θ)cos(π−θ)=sinθ=−cosθ

Symmetry About the Origin

SL 3.5

sin(θ+π)cos(θ+π)=−sinθ=−cosθ

Relating Angles Between Quadrants

SL 3.5

Once the values of sin and cos are memorized for critical angles in the first quadrant, you can use the rules of symmetry and periodicity to determine the values of critical angles in arbitrary quadrants.