SOLs addressed:

T.1             The student will use the definitions of the six trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of an angle in standard position, given a point, other than the origin, on the terminal side of the angle. Circular function definitions will be connected with trigonometric function definitions.

T.2             The student, given the value of one trigonometric function, will find the values of the other trigonometric functions. Properties of the unit circle and definitions of circular functions will be applied.

T.3             The student will find without the aid of a calculating utility the values of the trigonometric functions of the special angles and their related angles as found in the unit circle. This will include converting radians to degrees and vice versa.

In this activity, students will use Geometer's Sketchpad to manipulate the unit circle and use it to determine the values of the trigonometric functions of special angles.  Students will discover the relationship between polar and rectangular coordinates and how to apply these relationships using the Pythagorean Theorem and Special Triangles.

If you have Geometer's Sketchpad, click here to download your unit circle.  If not, follow these directions to create your own.

Directions:

• Open GSP to a blank page.
• Under the Grid menu, select Define Coordinate System
• Using the point at (0,0) and the point at (1,0), construct a circle with radius 1 centered at the origin
• With the circle still highlighted, under the Grid menu, select Define Unit Circle (if asked if you want to define another coordinate system, click yes)
• By clicking on the point at (1,0) and dragging, enlarge the unit circle so that it almost fills your screen
• Create a segment from the origin to a point on the circle in quadrant I
• Create a segment on the x-axis from the point (1,0) to (-1,0)
• Find the measure of the angle you have created by selecting (in order) point (1,0), the origin, the other point on your unit circle; then under the measure menu, select Angle
• Try dragging point C around the circle in the first and second quadrants; Does your angle measure increase?  Where does it measure 90 degrees?
• With point C in the first quadrant, create a line perpendicular to the x-axis through that point.  What happens when you drag point C past 90 degrees?
• To correct this problem, place point C in the second quadrant and create a line perpendicular to the x-axis through point C
• Now, we will track the coordinate of point C:  Select point C; under the Measure menu, select Coordinates
• What should happen to the x-coordinate as you drag point C past 90 degrees?
• What should be the length of segment BC?
• Check your answer by selecting segment BC; under Measure, select Coordinate distance.  The distance should be 1.  What is another name for this segment in relation to the circle?

Now, you have created your unit circle.  To find the values of the special angles, use the Excel chart attached here and answer the following questions:

1)  Find the point on the unit circle in the first quadrant where the x and y coordinates are the same.  What is the degree measure of angle ABC at this point?  Enter this degree measure under the DEGREE column of your Excel spreadsheet.  The RADIANS column gives you the multiple of π that corresponds to the angle in degrees (i.e. 1/6 means that the radian measure is π/6).

2)  What are the coordinates of a 60 degree angle?  a 30 degree angle?  Enter these angles under the DEGREE column in your Excel spreadsheet.  Try out a few other angles.

3)  What is are the coordinate of the zero degree angle?  Enter this angle in your Excel spreadsheet.  What are the values of sin 0 and cos 0?  Which of these corresponds to the x coordinate of the zero degree angle?  the y coordinate?

4)  What is the coordinate of a 90degree angle?  Enter this angle in your Excel spreadsheet.  What are the values of sin 90  and cos 90?  Which of these corresponds to the x coordinate?  the y coordinate?

5)  Based on your answers for questions (3) and (4), which rectangular coordinate can be written as sin θ?  Which can be written as cos θ? 

6)  Using your conclusions from (5) and the following reference chart, determine the exact measure of the legs of a 45-45-90 triangle and a 30-60-90 triangle. 

√(2)/2	0.707107
√(3)/2	0.866025
√(3)	1.732051

Draw these two triangles.  Verify the lengths of the sides using the Pythagorean Theorem.

7)  Now, complete the following chart using the exact values for sin θ, cos θ, tan θ, etc.

Degrees	Radians	Sin θ	Cos θ	Tan θ	Csc θ	Sec θ	Cot θ
0
30
45
60
90
180
360