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Finding Trigonometric Functions Using 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, as shown in (Figure). 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]

The x- and y-axes divide the coordinate plane 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 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)=\mathrm{cos}\,t\,[/latex]and[latex]\,f\left(t\right)=\mathrm{sin}\,t,[/latex]respectively. This means[latex]\phantom{\rule{0.3em}{0ex}}x=\text{cos }t\phantom{\rule{0.3em}{0ex}}[/latex]and[latex]\phantom{\rule{0.3em}{0ex}}y=\text{sin }t.[/latex]

Defining Sine and Cosine Functions from the Unit Circle

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), 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), 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]\,\mathrm{sin}\,t\,[/latex]is the same as[latex]\,\mathrm{sin}\left(t\right)\,[/latex]and[latex]\,\mathrm{cos}t\,[/latex]is the same as[latex]\,\mathrm{cos}\left(t\right).\,[/latex]Likewise,[latex]\,{\mathrm{cos}}^{2}t\,[/latex]is a commonly used shorthand notation for[latex]\,{\left(\mathrm{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.

Sine and Cosine Functions

If[latex]\,t\,[/latex]is a real number and a point[latex]\,\left(x,y\right)\,[/latex]on the unit circle corresponds to a central angle[latex]\,t,[/latex]then

[latex]\mathrm{cos}\,t=x[/latex]

[latex]\mathrm{sin}\,t=y[/latex]

How To

Given a point P[latex]\,\left(x,y\right)\,[/latex]on the unit circle corresponding to an angle of[latex]\,t,[/latex]find the sine and cosine.

1. The sine of[latex]\,t\,[/latex]is equal to the y-coordinate of point[latex]\,P:\text{sin }t\text{ = }y.[/latex]
2. The cosine of[latex]\,t\,[/latex]is equal to the x-coordinate of point[latex]\,P:\text{cos} t=x.[/latex]

Finding Function Values for Sine and Cosine

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

Show Solution

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

[latex]\begin{array}{ccc}\hfill x& =\mathrm{cos}\,t\hfill & =\frac{1}{2}\hfill \\ y& =\mathrm{sin}\,t\hfill & =\frac{\sqrt{3}}{2}\hfill \end{array}[/latex]

Try It

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). Find[latex]\,\mathrm{cos}\,t\,[/latex]and[latex]\,\mathrm{sin}\,t.[/latex]

Show Solution

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

Calculating Sines and Cosines along an Axis

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

Show Solution

Moving[latex]\,90°\,[/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[latex]\,\left(0,1\right),[/latex]as shown in (Figure).

We can then use our definitions of cosine and sine.

[latex]\begin{array}{cccc}\hfill x& =\text{cos }t\hfill & =\mathrm{cos}\left(90°\right)\hfill & =0\hfill \\ \hfill y& =\text{sin }t\hfill & =\mathrm{sin}\left(90°\right)\hfill & =1\hfill \end{array}[/latex]

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

The Pythagorean Identity

Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is[latex]\,{x}^{2}+{y}^{2}=1.\,[/latex]Because[latex]\,x=\mathrm{cos}\,t\,[/latex]and[latex]\,y=\mathrm{sin}\,t,[/latex]we can substitute for[latex]\,x\,[/latex]and[latex]\,y\,[/latex]to get[latex]\,{\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t=1.\,[/latex]This equation,[latex]\,{\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t=1,[/latex]is known as the Pythagorean Identity. See (Figure).

We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution.

Finding a Cosine from a Sine or a Sine from a Cosine

If[latex]\,\mathrm{sin}\left(t\right)=\frac{3}{7}\,[/latex]and[latex]\,t\,[/latex]is in the second quadrant, find[latex]\,\mathrm{cos}\left(t\right).[/latex]

Show Solution

If we drop a vertical line from the point on the unit circle corresponding to[latex]\,t,[/latex]we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. See (Figure).

Substituting the known value for sine into the Pythagorean Identity,

[latex]\begin{array}{ccc}\hfill {\text{cos}}^{2}\left(t\right)+{\mathrm{sin}}^{2}\left(t\right)& =& 1\hfill \\ \hfill {\text{cos}}^{2}\left(t\right)+\frac{9}{49}& =& 1\hfill \\ \hfill {\text{cos}}^{2}\left(t\right)& =& \frac{40}{49}\hfill \\ \hfill \text{cos}\left(t\right)& =& ±\sqrt{\frac{40}{49}}=±\frac{\sqrt{40}}{7}=±\frac{2\sqrt{10}}{7}\hfill \end{array}[/latex]

Because the angle is in the second quadrant, we know the x-value is a negative real number, so the cosine is also negative.

[latex]\text{cos}\left(t\right)=-\frac{2\sqrt{10}}{7}[/latex]