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The Unit Circle
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The unit circle: using \(\cos\theta\) and \(\sin\theta\) as the \(x\)- and \(y\)-coordinates of a point on the circle, key exact values and quadrants, symmetry rules, periodicity, tangent as \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and \(\sin^2\theta+\cos^2\theta=1\).
Want a deeper conceptual understanding? Try our interactive lesson!
Sine and Cosine on the Unit Circle
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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
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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
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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
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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
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sin(−θ)cos(−θ)=−sinθ=cosθ
Symmetry About the Y-axis
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sin(π−θ)cos(π−θ)=sinθ=−cosθ
Symmetry About the Origin
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sin(θ+π)cos(θ+π)=−sinθ=−cosθ
Relating Angles Between Quadrants
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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.
Nice work completing The Unit Circle, here's a quick recap of what we covered:
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