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Handout Tuesday 9/23
The goal of section 2.2 is to develop rules for the derivatives of \(\sin(x)\) and \(\cos(x)\text{.}\) We could think of these rules as arbitrary, but to help us remember and make sense of them, it will be useful to briefly review some trigonometry.
Perhaps you learned trigonometry as a method for finding the lengths of legs of right triangles. What does \(\sin(x)\) and \(\cos(x)\) tell you there?
We want to extend this meaning to a larger domain of angles, not just between 0 and 90 degrees.
Now we want the triangle with base on the \(x\)-axis and angle at the origin to still obey the usual relationships for \(\sin(\theta)\) and \(\cos(\theta)\text{.}\) So this means that \(\sin(\theta)\) will be the \(y\)-value of the circle, and \(\cos(\theta)\) is the \(x\)-value.
Now continuing this, it makes sense to ask for \(\sin(3\pi/4)\text{.}\) We just need to know what the \(y\) value is on the circle at that point.
Notice that we are using a circle with radius 1 here, so while \(\sin(\theta) = y/r\text{,}\) we really get \(y\text{.}\)
