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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.

Sketch

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 sin t = y/rand cos t = x/r.

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 radius 1.

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Investigate

  1. 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?
  2. 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?
  3. 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.
  4. 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.
    1. sin 150°
    2. cos 120°
    3. cos 225°
  5. In what quadrants is sin t positive? Negative?
  6. In what quadrants is cos t positive? Negative?
  7. What are the largest and smallest values of sin t?
  8. What are the largest and smallest values of cos t?
  9. When is sin t equal to 0? When is cos t equal to 0?
  10. Suppose sin t is approximately –0.731, and t is between 180° and 270°. What is t?
  11. 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 your angles?

Sketch

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 the animation. Press Start Over to return everything to the starting positions, and click X to erase the traces.

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Investigate

  1. In theory, what is the domain of the sine function?
  2. What is the range of the sine function?
  3. 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?

Sketch

This sketch is similar to the previous sketch but it plots the function f(t) = cos t.

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Investigate

  1. What is the domain of the cosine function?
  2. What is the range of the cosine function?
  3. What is the period of the cosine function?
  4. Explain why the period of the sine function is the same as the period of the cosine function?
  5. Describe similarities between the plots of the two functions.
  6. How could you translate the plot of the sine function to be congruent to the plot of the cosine function?