Defining the Circular Functions*On this web page you can explore the definitions and graphs of the
circular functions sine and cosine. The sketches on this page will give
you a deeper understanding of Lesson 10.1 of Discovering
Advanced Algebra: An Investigative Approach.
The sketch below shows a circle with radius r and center at
the origin. A central angle of t degrees is shown in standard
position, measured counterclockwise from the positive x-axis
to the terminal sider. From your knowledge of right
triangle trigonometry, you can see that and .
You can use this sketch to define sine and cosine for all angles,
not just those between 0° and 90°. To determine the sine or
cosine of angle t, use the reference angle—the acute
angle between the terminal side and the x-axis. The reference
triangle is a right triangle that contains the reference angle,
colored purple in this sketch.
Drag point (x, y) to change the angle. Drag the point
labeled Change Radius to
change the radius of the circle, or press Unit Circle to make the
- Drag (x, y) around the circle and observe the
measures of the central angle and the reference angle. How are they
related for the different quadrants?
- For any position of (x, y), change the radius of
the circle. Do the values of sin t and cos t change?
Why or why not?
- Press Unit Circle and
then drag (x, y) around the
circle. What do you notice about the coordinates of (x, y)
and the values of sin t and cos t in the unit circle?
Explain why this happens.
- Use the sketch to approximate these values. For each, draw the
reference triangle on your own paper and explain how the measurements
of the triangle contribute to your answer.
- sin 150°
- cos 120°
- cos 225°
- In what quadrants is sin t positive? Negative?
- In what quadrants is cos t positive? Negative?
- What are the largest and smallest values of sin t?
- What are the largest and smallest values of cos t?
- When is sin t equal to 0? When is cos t equal to
- Suppose sin t is approximately –0.731, and t is
between 180° and 270°. What is t?
- Find all values of t between 0° and 360° for
which cos t is approximately 0.616. In which quadrant is each
angle located? What's true about the reference triangles for all of
On the left side of this sketch you see a unit circle with a
central angle of t degrees. On the right is a coordinate system
that plots the function f(t) = sin t. Press Show Sine Function to
animate the point (x, y) around
the circle and trace the sine function. Press the button again to stop
animation. Press Start Over
to return everything to the starting positions,
and click X to erase the traces.
- In theory, what is the domain of the sine function?
- What is the range of the sine function?
- The sine function is periodic because the output values
of the function repeat at regular intervals. The period of a
function is the smallest distance between values of the independent
variable before the cycle begins to repeat. Watch the animation
carefully to determine when the sine function begins to repeat. What is
the period of the sine function?
This sketch is similar to the previous sketch but it plots the
function f(t) = cos t.
- What is the domain of the cosine function?
- What is the range of the cosine function?
- What is the period of the cosine function?
- Explain why the period of the sine function is the same as the
period of the cosine function?
- Describe similarities between the plots of the two functions.
- How could you translate the plot of the sine function to be
congruent to the plot of the cosine function?