Applying Trig Functions to Angles of Rotation

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Trigonometry - Keys to remember the 6 Basic Trigonometric Functions

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Key Questions

  • Any angle of rotation #theta# can be represented by a point #A# on a unit circle with a center at the origin of coordinates #O# and radius #1#. The angle is measured counterclockwise from the positive direction of the X-axis to a line from #O# to #A#, so #angle XOA=theta# with #|OA|=1#. Thus, an angle of #90^0# is represented by a point with coordinates #(0,1)#, an angle of #270^0# is represented by a point #(0,-1)# etc.

    Then, by definition, if point #A# has coordinates #(A_x,A_y)#,
    #sin(theta)=A_y#
    #cos(theta)=A_x#
    #tan(theta)=A_y/A_x# (for #A_x!=0#)
    #sec(theta)=1/A_x# (for #A_x!=0#)
    #csc(theta)=1/A_y# (for #A_y!=0#)

    The above are definitions of trigonometric functions for any angles. The typical geometric definition of trigonometric functions using the right triangles is not general enough, while the above definitions work for all angles and, in case of acute angles in the right triangles, are identical to geometric definition.

    I might suggest to study trigonometry at Unizor - Trigonometry. The site has a very detailed explanation of properties of trigonometric functions based on the above definition.

  • We use two special right triangles - 45-45-90 and 30-60-90 - and draw them in each quadrant to create the coordinates on the unit circle.

    Let's start with what the special right triangles look like.

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    We make the hypotenuse (radius) 1 so that each triangle will fit within the same circle.

    Now, take the 45-degree reference angle triangle and draw it in each quadrant.
    enter image source here

    The adjacent leg represents the x-value and the opposite leg represents the y-value. This creates the four coordinates for the 45-45-90 triangle (#pi/4, (3pi)/4, (5pi)/4, (7pi)/4#).

    Let's move on to the 30-degree reference angle triangles.
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    This produces the coordinates for #pi/6, (5pi)/6, (7pi)/6, and (11pi)/6#.

    Finally, let's draw the 60-degree reference angle triangles.

    enter image source here

    This produces the coordinates for #pi/3, (2pi)/3, (4pi)/3, and (5pi)/3#.

  • 1st and 3rd quadrants. You can use the CAST diagram which tells you in which quadrants angles are positive. ie. cos (and its reciprocal sec) is + in the4th quad, All of them + in 1st quad, sin (and its reciprocal cosec) + in 2nd and tan and cot + in 3rd.

  • The unit circle simply put is a circle with a radius of 1. Whether this may be 1 inch, meter or mile it doesn't matter as long as the value is 1.

    The unit circle has a distance of #2pi# around the circumference. The value #2pi# is almost the same as #360^o#. The difference is that #2pi# is the DISTANCE around the circle with a radius of 1, whereas #360^o# is the ANGLE of a circle with ANY given radius.

    Here's an example, if you walked in a perfect circle that has a radius of 1 mile, in total you would have walked #2pi# miles, which is equivalent to 6.28 miles

    Hope this helps!

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