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Geometry/Unit Circle


Hello, I am confused about how to solve some problems on unit circle. Can you please explain how to simplify these. These are a few of the problems.

1) cos² (11π/24)+ sin²(11π/24)
*also for this one I don't know what to do with the ^2
2) tan (π/4)
3) cos (π/3)+ sin (π/6)

Hi J.D.,

Solving problems on the unit circle isn't too bad once you understand what's going on.

On a Cartesian plane, the unit circle is a circle of radius 1 centered at the origin.  All points on the circle have coordinates (x,y)=(cos t, sin t), where t is the angle measured to the positive x-axis.

Draw a straight line from the origin to any point on the circle.  From that point on the circle, draw a vertical line to the x-axis.  You will now have a right triangle.  The height will be cos t, and the base sin t, where t is the angle between the positive x-axis and the radius you initially drew.  That radius is the hypotenuse of the right triangle.  Notation-wise, cos²(t) means (cos t)².  Therefore, by the Theorem of Pythagoras, cos²(t)+sin²(t)=1².  Note that the particular angle you chose was not important: the above is true for any angle t.

There are 2π radians (or 360°) in a circle.  The important radian angles are 0, π/6, π/4, π/3, and π/2 (0°, 30°, 45°, 60°, and 90°, respectively).  These all lie in the first quadrant, and you can use symmetry to figure it out for the others.  Do a Google image search for unit circle and you'll find some good diagrams with all that you need to know.  Once you know these, it's generally just a matter of plugging some numbers in.  Once you remember the pairs (1,0), (1/2,√3/2), and (√2/2,√2/2), symmetry will help you find the rest.

tan(t)=sin(t)/cos(t) (you can consider this a definition for now, if you haven't seen an explanation), so

For the third question, again refer to the numbers on the diagram (which you will know by heart with practice).

Hope this has been helpful,


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Azeem Hussain


I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.


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