We all know the identities SOH-CAH-TOA. [br][br][math]sin\left(x\right)=\frac{o}{h}[/math][br][math]cos\left(x\right)=\frac{a}{h}[/math][br][math]tan\left(x\right)=\frac{o}{a}[/math][br][br]But how do these come to be?[br]Let's first take a look at the unit circle.

How did we create the functions sine and cosine?[br][br]To figure this out let take a look at the two unit circles.[br]How is the length of BD related to the y-coordinate of B? related to the sine of a (shown at bottom of legend)?[br][br]How is the length of AD related to the x-coordinate of B? related to the cosine of a (shown at bottom of legend)?

Think about these given circles.

What if we wanted sine/cosine in radii?[br][br]To get sine in terms of radii, we would have to convert the length of sine (BC) to radii.[br]You can do a similar thing with cosine.[br]What is the relationship between this length and the sine/cosine of the angle.[br][br]So in order to find sine/cosine you have to divide the vertical/horizontal height (opposite/adjacent) by the radius. But hey! the hypotenuse of the triangle is equal to the radius. This is how we get:[br][br][img]data:image/png;base64,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[/img][br][img]data:image/png;base64,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[/img][br][br]How can we extend this to tangent?[br][br]The tangent happens to be the slope of line AB. Using the slope formula, we can see, if:[br][math]B=\left(a,o\right)[/math][br]then the the slope of AB is:[br][math]\frac{o-0}{a-0}=\frac{o}{a}[/math][br][br][br]