Special Angles in the Unit Circle
How to use the symmetry of a unit circle to find values of sine and cosine functions to the angles related by symmetry to the special angles π/6, π/4 and π/3?
Special Angles in the Unit Circle Any line through the center of a circle is a line of symmetry. The center of the circle is point of symmetry. We now consider a unit circle with center at the origin of a system of x and y axes. We are here interested in the symmetries with respect to the origin, to the xaxis and to the yaxis. Knowing the values of the sine and cosine of the angles in the first quadrant, it is easier to use the symmetry of the unit circle to obtain the sine and cosine of the angles in the other quadrants.
The unit circle below shows the values of the cosine and sine functions (coordinates in blue, with the xcoordinate being the cosine and the ycoordinate is the sine) for the special angles:
. Question: How to find the sine and cosine of an angle between 0 and 2π related by symmetry to any of the angles π/6, π/4 or π/3?
Example 1: Consider the angles: 2π/3, 4π/3 and 5π/3. They all have a relationship with π/3:
Example 2: Consider the angles: 5π/6, 7π/6 and 11π/6. They all have a relationship with π/6:
Example 3: Consider the angles: 3π/4, 5π/4 and 7π/4. They all have a relationship with π/4:

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