Learning objective
Recall the unit circle, which we can use to evaluate exact trigonometric ratios for \left( \cos \theta, \sin \theta \right):
The graph of trigonometric functions has the angles in the unit circle to represent the x-axis. For example:
Degree | 90\degree | 180\degree | 270\degree | 360\degree |
---|---|---|---|---|
Radian | \dfrac{\pi}{2} | \pi | \dfrac{3\pi}{2} | 2\pi |
The coordinates in the unit circle that ranges from -1 to 1 is used to represent the y-axis. For example:
Decimals | -1 | -0.5 | 0 | 0.5 | 1 |
---|---|---|---|---|---|
Fractions | -1 | -\dfrac12 | 0 | \dfrac12 | 1 |
Explore the applet by moving the slider.
Recall that sine is represented by the vertical leg of the right triangle positioned in the unit circle, or the y-coordinate.
Plotting the corresponding values of \sin \theta will result to the following graph.
Notice that as we move through values of \theta, the graph of f \left( \theta \right)= \sin \theta will oscillate accordingly between -1 and 1.
Consider the function f \left( \theta \right) = \sin \theta.
Complete the table with values in exact form:
\theta | 0 | \dfrac{\pi}{6} | \dfrac{\pi}{2} | \dfrac{5 \pi}{6} | \pi | \dfrac{7 \pi}{6} | \dfrac{3 \pi}{2} | \dfrac{11 \pi}{6} | 2 \pi |
---|---|---|---|---|---|---|---|---|---|
\sin \theta |
Sketch a graph for f \left( \theta \right) = \sin \theta on the domain [-2\pi, 2\pi].
State the sign of \sin \left( \dfrac{- \pi}{12} \right).
The graph of f \left( \theta \right) = \sin \theta relates closely to the y-coordinate of points on the unit circle.
Explore the applet by moving the slider.
Recall that cosine is represented by the horizontal leg of the right triangle positioned in the unit circle, or the x-coordinate.
Plotting the corresponding values of \cos \theta will result to the following graph.
As we move through values of \theta, the graph of f \left( \theta \right) = \cos \theta will oscillate accordingly between -1 and 1.
Consider the function f \left(\theta \right) = \cos \theta.
Complete the table with values in exact form:
\theta | 0 | \dfrac{\pi}{3} | \dfrac{\pi}{2} | \dfrac{2 \pi}{3} | \pi | \dfrac{4 \pi}{3} | \dfrac{3 \pi}{2} | \dfrac{5 \pi}{3} | 2 \pi |
---|---|---|---|---|---|---|---|---|---|
\cos \theta |
Sketch a graph for f \left( \theta \right) = \cos \theta on the domain [-2 \pi, 2 \pi].
State the sign of \cos \left( \dfrac{-\pi}{12} \right).
The graph of f \left( \theta \right) = \cos \theta relates closely to the x-coordinate of points on the unit circle.