To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1. The angle (in radians) that [latex]t[/latex] intercepts forms an arc of length [latex]s[/latex]. Using the formula [latex]s=rt[/latex], and knowing that [latex]r=1[/latex], we see that for a unit circle, [latex]s=t[/latex].

For any angle [latex]t[/latex], we can label the intersection of the terminal side and the unit circle as by its coordinates, [latex]\left(x,y\right)[/latex]. The coordinates [latex]x[/latex] and [latex]y[/latex] will be the outputs of the trigonometric functions [latex]f\left(t\right)=\cos t[/latex] and [latex]f\left(t\right)=\sin t[/latex], respectively. This means [latex]x=\cos t[/latex] and [latex]y=\sin t[/latex].

Defining Sine and Cosine Functions

Now that we have our unit circle labeled, we can learn how the [latex]\left(x,y\right)[/latex] coordinates relate to the arc length and angle. The sine function relates a real number [latex]t[/latex] to the y-coordinate of the point where the corresponding angle intercepts the unit circle. More precisely, the sine of an angle [latex]t[/latex] equals the y-value of the endpoint on the unit circle of an arc of length [latex]t[/latex]. In Figure 2, the sine is equal to [latex]y[/latex]. Like all functions, the sine function has an input and an output. Its input is the measure of the angle; its output is the y-coordinate of the corresponding point on the unit circle.

The cosine function of an angle [latex]t[/latex] equals the x-value of the endpoint on the unit circle of an arc of length [latex]t[/latex]. In Figure 3, the cosine is equal to [latex]x[/latex].

Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: [latex]\sin t[/latex] is the same as [latex]\sin \left(t\right)[/latex] and [latex]\cos t[/latex] is the same as [latex]\cos \left(t\right)[/latex]. Likewise, [latex]{\cos }^{2}t[/latex] is a commonly used shorthand notation for [latex]{\left(\cos \left(t\right)\right)}^{2}[/latex]. Be aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra parentheses when entering calculations into a calculator or computer.

Point [latex]P[/latex] is a point on the unit circle corresponding to an angle of [latex]t[/latex]. Find [latex]\cos \left(t\right)[/latex] and [latex]\text{sin}\left(t\right)[/latex].

Show Solution

We know that [latex]\cos t[/latex] is the x-coordinate of the corresponding point on the unit circle and [latex]\sin t[/latex] is the y-coordinate of the corresponding point on the unit circle. So:

[latex]\begin{align} x&=\cos t=\frac{1}{2} \\ y&=\sin t=\frac{\sqrt{3}}{2} \end{align}[/latex]

A certain angle [latex]t[/latex] corresponds to a point on the unit circle at [latex]\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)[/latex] as shown in Figure 5. Find [latex]\cos t[/latex] and [latex]\sin t[/latex].

Show Solution

[latex]\cos \left(t\right)=-\frac{\sqrt{2}}{2},\sin \left(t\right)=\frac{\sqrt{2}}{2}[/latex]

Finding Sines and Cosines of Angles on an Axis

For quadrantral angles, the corresponding point on the unit circle falls on the x- or y-axis. In that case, we can easily calculate cosine and sine from the values of [latex]x[/latex] and [latex]y[/latex].

Find [latex]\cos \left(90^\circ \right)[/latex] and [latex]\text{sin}\left(90^\circ \right)[/latex].

Show Solution

Moving [latex]90^\circ[/latex] counterclockwise around the unit circle from the positive x-axis brings us to the top of the circle, where the [latex]\left(x,y\right)[/latex] coordinates are (0, 1).
Using our definitions of cosine and sine,

[latex]\begin{align}x&=\cos t=\cos \left(90^\circ \right)=0\\ y&=\sin t=\sin \left(90^\circ \right)=1\end{align}[/latex]

The cosine of [latex]90^\circ[/latex] is [latex]0[/latex]; the sine of [latex]90^\circ[/latex] is [latex]1[/latex].