Recall the definitions of the 6 trigonometric functions defined at an angle drawn in standard position within the coordinate plane. (These ratios were defined in terms of [i]x[/i], [i]y, [/i]& [i]r[/i]). [br][br]Interact with this diagram for a minute or two. (The 2 LARGE POINTS are moveable). [br]Then, answer the question prompts that follow.
Explain why each segment IS what it is. (Some are much easier than others). [br][br]For example, how do we know the [color=#9900ff][b]purple segment has a length = to the tangent of [math]\theta[/math][/b][/color]?
How many pairs of similar triangles do you see here? How do we know these triangles you reference are all similar to each other? ([color=#cc0000][b]Hint: Slide the red slider for some insight![/b][/color])
You have previously learned that similar triangles have corresponding sides that are in proportion. That is, ratios of corresponding sides of similar triangles are all equal in value. [br][br]Given this fact, what other trig identities can we author from this picture?
Example: [br][br]Since the radius of this circle = 1, by the Pythagorean Theorem, we can write[br][math]\left(\sin\left(\theta\right)\right)^2+\left(\cos\left(\theta\right)\right)^2=1[/math]. [br][br]Granted, we didn't need to use the fact that the triangles were similar to conclude this. [br]Yet MANY NEW TRIG ID's can be authored by using the fact that we have similar triangles here. [br][br][b]What other relationships do you see here? [/b]