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Question
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)

Answer
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
tan(π/4)
=sin(π/4)/cos(π/4)
=(√2/2)/(√2/2)
=1

For the third question, again refer to the numbers on the diagram (which you will know by heart with practice).
cos(π/3)+sin(π/6)
=(1/2)+(1/2)
=1

Hope this has been helpful,
Azeem

Geometry

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Azeem Hussain

Expertise

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

Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

Publications
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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