Understanding Trigonometric Functions

The applet below will help students understand why the sine and cosine function behave and appear the way they do.

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Instructions:

The circle given within the applet is called the unit circle where r=1. A right triangle has been formed within the circle, move point C around the circle to ensure that at all places you have a right triangle.

T or F: ALL right triangles have a similar triangle that resides in this construction, i.e. in the unit circle. Explain your reasoning.

The Sine Function has been defined in class to be the ratio between the Length of the Opposite side of an angle / the Length of the Hypotenuse of the triangle. Pull out our calculator and verify the sine of four different points by using the text in the upper part of the screen. Be sure to make all of the points be in different quandrants of the circle.

Were all your answers the same as the SINE(A) text? What were the differences? Why?

The Graph of the Sine Function is a wave, click on Show SINE Curve to show it. How do you think we can get that shape?

Is there a relationship with the y-value of C and the y-values of the Sine Curve?

The Sine Curve is defined to be (measure of the Angle A in radians, y-value of C) Click on Show SINE Point. Move C around the circle again and notice if the red dot stays on the Sine Curve or not.

Deselect both check marks.

The Cosine Function has been defined in class to be the ratio between the Length of the Adjacent side of an angle / the Length of the Hypotenuse of the triangle. Pull out our calculator and verify the cosine of four different points by using the text in the upper part of the screen. Be sure to make all of the points be in different quandrants of the circle.

Were all your answers the same as the COSINE(A) text? What were the differences? Why?

The Graph of the Cosine Function is a wave, click on Show COSINE Curve to show it. How do you think we can get that shape?

Is there a relationship with the x-value of C and the y-values of the Cosine Curve?

The Cosine Curve is defined to be (measure of the Angle A in radians, x-value of C) Click on Show COSINE Point. Move C around the circle again and notice if the blue dot stays on the Coine Curve or not.

Bryan Stephenson, Created with GeoGebra