Kristy Hubbs
Tech Project #1
GeoGebra Project #1
Trigonometric Ratios and the Unit Circle
You have learned that the trigonometric ratios are defined in terms of the sides of a right
triangle, which limits you to acute angles. In the coordinate plane, it’s possible to define
trigonometric ratios for angles with measures less than 0° and greater than 90° by using
what is called the unit circle. This is discussed and used in great detail in Pre-Calculus
and Calculus. The unit circle is a circle with center (0, 0) and radius 1 unit.
In this activity, you'll use GeoGebra to explore how a unit circle simplifies the
trigonometric ratios for acute angles. Then you'll use the unit circle to explore the ratios
for all angles, from 0° to 360°, and even negative angle measures. First open the
Trig_and_Unit_Circle2 file.
Step 1 - Write the ratios for sine, cosine and tangent of <DAC in terms of the sides of
triangle ADC.
Step 2 - What is the radius of this circle? How does that change your trigonometric
definitions from step 1? Write the new simplified trigonometric definitions.
Step 3 - What is the coordinate of C? Which value corresponds to the sine of <DAC?
Cosine <DAC? What physical part of the sketch represents these values? (click on the
first checkbox and check your results against those on the screen)
Step 4 - What is the ratio of the y-coordinate of C to the x-coordinate of C? Does this
equal anything familiar? What is it? (click on the checkbox two. Does your result
match?)
Step 5 - Click on the third checkbox. What are the coordinates of point E? Which value
corresponds to the tangent? What part of the sketch represents the tangent? Use similar
triangles to explain your answer.
If point C is in another quadrant, <DAC is still acute. To modify the definition of the
trigonometric ratios, you need to measure the rotation from AB to AC. You do this by
measuring the arc from B to C, which is equal to the measure of its central angle
(<DAC).
Step 6 - Click on the fourth checkbox. Drag point C around the circle and watch what
happens to arc BC. Summarize your observation.
Step 7 - Drag point C around the circle again. This time watch how the trigonometric
ratios change for angle measures between 0° and 360°. Answer the following question.
a) How does the sine change as the arc goes from 0° to 90° to 180°? From 180° to 270° to
360°?
b) How does the cosine change over the same intervals?
c) How does the tangent change on those intervals? What happens as the tangent
approaches 90°? 270°?
What do you think is the value of the tangent of 90°?
Additional Questions:
1) How do you think you could get a negative angle measure from this unit circle?
2) How would this affect the trigonometric ratios?
3) What are the signs of the trigonometric ratios in each quadrant?
Prepare an organized presentation of your observations and results from this activity. Be
sure to include your answers to all questions and any sketches that you feel necessary.