Special Triangles and Trigonometry
Triangle ABC below is a 30-60-90 triangle since it's angle measures are 30º,60º, and 90º. The sides have been measured for you, as well as the ratios between the sides.
What happens to the ratio between side AB and side AC as you move the vertices of the triangle? Why do you think this happens?
What happens to the ratio between side BC and side AC as you move the vertices of the triangle? Why do you think this happens?
Make leg AB of the triangle have length 1, what are the [i]EXACT[/i] lengths of the other two legs?[br]Describe how you found the exact lengths.
Triangle ABC below is a 45-45-90 triangle since it's angle measures are 45º, and 45º. The sides have been measured for you, as well as the ratios between the sides.
What happens to the ratio between side AB and side AC as you move the vertices of the triangle? Why do you think this happens?[br]
Make leg AB of the triangle have length 1, what are the [i]EXACT[/i] lengths of the other two legs?[br]Describe how you found the exact lengths.
Chance for Review
Why is it that for the 30-60-90 triangle, we had two ratios, and for the 45-45-90 triangle only one ratio was described? (Hint: Think about the side lengths.)
*Be sure to fill our your note sheet*
When you are done, write your observations on the note sheet provided for you. (Hint: Instead of having length 1, one of the legs of the triangle has length x. How does this change the ratios?)
Using the characteristics you discovered in the previous activity, without measuring the sides, what are the exact measures of each of the missing sides for the following triangles?
(Note: If it helps, draw out the triangles on paper or use the previous activity)
Missing Side lengths of Special triangles
Use the Space below to write down the length of the missing sides. (Write in the form KM=# for each side)
Describe how you found each of the missing sides. What information did you use and why? Is there a way to check if your side lengths are correct (Without measuring)?
Trigonometry and Special Triangles!
These triangles (30-60-90 and 45-45-90) are called special triangles because of the special ratio between the sides. The angles that make up these triangles are often referred to as [i]special angles [/i]because they are very easy to represent trigonometric angles with. [br][br]The three trig functions we are going to start with are: [br] - Sine -Cosine -Tangent[br]
Use SOH CAH TOA to fill in the table create below using the triangles for each of the trig functions. (Don't input number values but rather exact ratios, e.g. "=AB/AC"
After creating a table of the different trig functions for each of the different angles, what happens to the ratios as you move the vertices of the two triangles? Why do you think this is the case?
How do the values in the table relate to the ratios between the side lengths of the special triangles from the first activity?
****Class Discussion and Note Sheet*****
Consider making the length of the hypotenuse of each of the special triangles to be 1.[br]What would the exact length of each of the legs be?[br]Using our tables, what would the decimal approximation of each of the trig functions be?[br]Do these values in the table match up with the exact values for the trig functions that we can find with SOH CAH TOA?[br](Fill in the provided triangles on your note sheet.)
Observe the Unit Circle Below
Move the slider on the unit circle to create the special triangle. How does the length of the leg opposite the measured angle relate to the trig values we found? [br]
Is there a specific trig function that this side could represent? What is it? Describe your reasoning.
*****Class Discussion***** Graph of Sine
What are some things that you notice about the graph of sine?
How would you describe the difference in graphing cosine instead of sine?[br](In other words, which leg of the triangle would be used to help graph cosine and why?)