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The Unit Circle
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The unit circle: using cosθ and sinθ as the x- and y-coordinates of a point on the circle, key exact values and quadrants, symmetry rules, periodicity, tangent as tanθ=cosθsinθ, and sin2θ+cos2θ=1.
Want a deeper conceptual understanding? Try our interactive lesson!
Sine and Cosine on the Unit Circle
AHL AI 3.8
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
AHL AI 3.8
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:
Quadrants
AHL AI 3.8
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.
Periodicity
SL AI 2.5
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
AHL AI 3.8
sin(−θ)cos(−θ)=−sinθ=cosθ
Symmetry About the Y-axis
AHL AI 3.8
sin(π−θ)cos(π−θ)=sinθ=−cosθ
Symmetry About the Origin
AHL AI 3.8
sin(θ+π)cos(θ+π)=−sinθ=−cosθ
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