RobotMath: Trigonometry on the Unit Circle

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In the previous lesson, RobotMath:: Solving Triangles, Part 1, we saw how the sine, cosine, and tangent functions, and their inverses, the arc functions, can be used to solve for properties of a right triangle. But that is only part of the story of these functions, since right triangles only contain angles between 0 and 90°. In this lesson we will generalize the definitions of the trigonometric functions to all angles, so that cases such as sin(-30°) or cos(225°) are well defined. This will allow us to solve new classes of problems in upcoming lessons.

The Unit Circle

Consider a circle centered on the origin. If the radius is 1, this is known as a unit circle. Every point on this circle has coordinates (x,y) such that x2+y2 = 1. Now consider one of these points that lies in the first quadrant, where x and y are both positive. Draw a line from the point (x,y) back to the origin (0,0), and another line from the point straight down to the x axis. Now we have a right triangle with an angle θ. Therefore, since r=1 we know that x=cos θ and y=sin θ.

[unit circle diagram]

Now consider a point in the second quadrant, as shown below. This point defines an angle of more than 90° relative to the positive x axis. We can measure the (x,y) coordinates of this point. and as before, we can take these as the values of cos θ and sin θ. Notice that now x and y are coordinates, not lengths, so they can be positive or negative, and in this case x is negative. Note also that dropping a perpendicular from (x,y) down to the negative x axis defines a right triangle with angle α that is supplementary to θ. So once again, sin θ and cos θ are defined as ratios with respect to the hypotenuse, whose length r = 1. The difference is that for angles outside of the first quadrant, θ is adjacent to rather than part of the right triangle, and the values of x and/or y may be negative.

[ unit circle with triangle in second quadrant]

Problems:

  1. Calculate sin(45°) and cos(45°) by drawing the angle on the unit circle and applying the Pythagorean theorem.
  2. Draw a diagram to show how to calculate sin(135°) and cos(135°), and derive the values.
  3. Draw a diagram to show how to calculate sin(225°) and cos(225°), and derive the values.