In the right triangle below, side [i]AC [/i]is fixed at 1 unit in length.[br]
Move point [i]B [/i]so that cosA is [math]\frac{1}{3}[/math].[br]What is the measure of angle [i]A[/i]?
Move point [i]B [/i]so that [i]cos [/i]A is [math]\frac{1}{10}[/math]. (You might have to zoom out.)[br]What is the measure of angle [i]A[/i]?
Play around with the above triangle and make [i]BC[/i] very big. What happens to [i]cos [/i]A as [i]BC [/i]gets bigger and bigger?
What happens to the measure of angle [i]A[/i] as [i]BC [/i]gets bigger and bigger?
Use the above exploration to speculate as to the value of [math]cos90^{\circ}[/math].
Use the above exploration to speculate as to the value of [math]sin90^\circ[/math].
Use the above exploration to speculate as to the value of [math]tan90^{\circ}[/math].
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In the right triangle above, what is [i]sin[/i] A?
a/c[br][math]\frac{a}{c}[/math]
In the right triangle above, what is [i]cos B[/i]?
a/c[br][math]\frac{a}{c}[/math]
Explain why [math]m\angle B=90^\circ-m\angle A[/math]
Because [i]sin A[/i] = [i]cos B[/i] and [i]B[/i] = 90 - [i]A[/i], we write [math]sinA=cos\left(90-A\right)[/math]. This is true for any angle [i]A. [/i]This is called a [i]cofunction[/i] identity.[br][br]Write another cofunction identity based on this idea.
Above is a 3-4-5 right triangle. Verify that the Pythagorean Theorem holds.
In the above right triangle, move point [i]B[/i] so that the hypotenuse has length 1. (You may need to zoom in a bit).
Verify that the Pythagorean Theorem holds in this triangle.
Notice that when the hypotenuse has length 1, we have [i]sin [/i]A = [i]BC [/i]and [i]cos [/i]A = [i]AC[/i]. Notice also that[br][br][math]\left(sinA\right)^2+\left(cosA\right)^2=1[/math]. This is also written as [math]sin^2A+cos^2A=1[/math].[br][br]This is called the [i]Pythagorean Identity[/i] and is true for any angle [i]A[/i].[br][br]
[b]BONUS[br][/b]In the above right triangle, verify the Pythagorean Identity. Do this by computing [i]sin[/i] A and [i]cos [/i]A and then showing that [br][br][math]sin^2A+cos^2A=1[/math].[br][br]Hint: You will need to use the relationship between [i]a[/i], [i]b[/i], and [i]c[/i] in the right triangle.