# Exploring Sine and Cosine of an Angle

## What's Behind the Sine and Cosine of an Acute Angle?

For the first activity, only slide the SINE slider to conceptually see the meaning of the Sine ratio within this right triangle context. Once you have completed all 6 activities below, repeat the process, but use the COSINE slider. Before you begin your work on the activities, spend a few minutes interacting with the applet. You can move the 2 LARGER WHITE VERTICES of the triangle anywhere you'd like. The black ACUTE ANGLE slider (or input box below it) is used to change the size of angle . Then see below for instructions.
EXPLORATORY QUESTIONS 1. What does the triangle look like when the sliders (Sine and acute angle) are set to the far left? Sketch what you see. Explain the relative sizes (hypotenuse, adjacent leg, and opposite leg). Now move the SINE slider to the far right. Write down what appears in the upper left corner. Explain the meaning of this. 2. Reset the SINE slider to the far left. Adjust the acute slider to 15 degrees. Sketch what you see. Now move the SINE slider to the far right. Sin(15 ) is approximately equal to what value? Explain what the value means in terms relative sizes of the legs. The _________ leg is ____% as long as the ____________. 3. Reset the SINE slider to the far left. Adjust the acute angle so that it's 30 degrees. Sketch what you see. Explain the numerical values and words in the upper left corner. The _________ leg is ____% as long as the ___________. 4. Reset the SINE slider to the far left. Set the acute angle to 45 degrees. Sketch what you see. Move the SINE slider to the right and explain what the Sin (45 ) means. The ______ leg is ___% as long as the ___________. 5. Reset the SINE slider to the far left. Set the acute angle so that it's 60 degrees. Sketch what you see. Move the SINE slider to the right and explain what the Sin ( ) means. The ______ leg is ___% as long as the ____________. 6. Reset the SINE slider to the far left. Set the acute angle to 85, 86, 87, 88, 89, and 90 degrees. Can you sketch what you see? Move the SINE slider to the right and explain what the Sin (90 ) means. The ______ leg is ___% as long as the ____________. Repeat the same 6 activities, BUT use COSINE instead of SINE.