Unit Circle Chart
Trigonometric Functions and the Unit Circle
We have already defined the trigonometric functions in terms of right triangles. In this section, we will redefine them in terms of the unit circle. Recall that a unit circle is a circle centered at the origin with radius 1. The angle [latex]t[/latex] (in radians ) forms an arc of length [latex]s[/latex].
The x- and y-axes divide the coordinate plane (and the unit circle, since it is centered at the origin) into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.
For any angle [latex]t[/latex], we can label the intersection of its side and the unit circle by its coordinates, [latex](x, y)[/latex]. The coordinates [latex]x[/latex] and [latex]y[/latex] will be the outputs of the trigonometric functions [latex]f(t) = \cos t[/latex] and [latex]f(t) = \sin t[/latex], respectively. This means:
[latex]\displaystyle{ \begin{align} x &= \cos t \\ y &= \sin t \end{align} }[/latex]
The diagram of the unit circle illustrates these coordinates.
We know that, for any point on a unit circle, the [latex]x[/latex]-coordinate is [latex]\cos t[/latex] and the [latex]y[/latex]-coordinate is [latex]\sin t[/latex]. Applying this, we can identify that [latex]\displaystyle{\cos t = -\frac{\sqrt2}{2}}[/latex] and [latex]\displaystyle{\sin t = -\frac{\sqrt2}{2}}[/latex] for the angle [latex]t[/latex] in the diagram.
Recall that [latex]\displaystyle{\tan t = \frac{\sin t}{\cos t}}[/latex]. Applying this formula, we can find the tangent of any angle identified by a unit circle as well. For the angle [latex]t[/latex] identified in the diagram of the unit circle showing the point [latex]\displaystyle{\left(-\frac{\sqrt2}{2}, \frac{\sqrt2}{2}\right)}[/latex], the tangent is:
[latex]\displaystyle{\begin{align}\tan t &= \frac{\sin t}{\cos t} \\&= \frac{-\frac{\sqrt2}{2}}{-\frac{\sqrt2}{2}} \\&= 1\end{align}}[/latex]
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