49 Unit Circle
Learning Objectives
In this section you will:
- Find function values for the sine and cosine of[latex]\,30°\text{ or }\left(\frac{\pi }{6}\right),45°\text{ or }\left(\frac{\pi }{4}\right),[/latex]and[latex]\,{60}^{\circ }\text{ or }\left(\frac{\pi }{3}\right).[/latex]
- Identify the domain and range of sine and cosine functions.
- Find reference angles.
- Use reference angles to evaluate trigonometric functions.
Looking for a thrill? Then consider a ride on the Singapore Flyer, the world’s tallest Ferris wheel. Located in Singapore, the Ferris wheel soars to a height of 541 feet—a little more than a tenth of a mile! Described as an observation wheel, riders enjoy spectacular views as they travel from the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a coordinate system. Then we can discuss circular motion in terms of the coordinate pairs.
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]
Unit Circle
A unit circle has a center at[latex]\,\left(0,0\right)\,[/latex]and radius[latex]\,1.\,[/latex]In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle[latex]\,t.[/latex]
Let[latex]\,\left(x,y\right)\,[/latex]be the endpoint on the unit circle of an arc of arc length[latex]\,s.\,[/latex]The[latex]\,\left(x,y\right)\,[/latex]coordinates of this point can be described as functions of the angle.
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,