Learning Objectives
In this section you will:
7.4.1 – Find function values for the sine and cosine using the unit circle.
7.4.2 – Find reference angles.
7.4.3 – 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.
7.4.1 – 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, 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
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.
- The sine of [latex]\,t\,[/latex] is equal to the y-coordinate of point [latex]\,P:\text{sin }t\text{ = }y.[/latex]
- The cosine of [latex]\,t\,[/latex] is equal to the x-coordinate of point [latex]\,P:\text{cos} t=x.[/latex]
Example 1 – 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]
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:
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 answer
[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]
Example 2 – Calculating Sines and Cosines along an Axis
Find [latex]\,\text{cos}\left(90°\right)\,[/latex] and [latex]\,\text{sin}\left(90°\right).[/latex]
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.
The cosine of [latex]\,90°\,[/latex] is 0; the sine of [latex]\,90°\,[/latex] is 1.
Try It
Find cosine and sine of the angle [latex]\,\pi .[/latex]
Show answer
[latex]\mathrm{cos}\left(\pi \right)=-1,\mathrm{sin}\left(\pi \right)=0[/latex]
Likewise, there will be an angle in the fourth quadrant with the same cosine as the original angle. The angle with the same cosine will share the same x-value but will have the opposite y-value. Therefore, its sine value will be the opposite of the original angle’s sine value.
As shown in (Figure), angle [latex]\,\alpha \,[/latex] has the same sine value as angle [latex]\,t;[/latex] the cosine values are opposites. Angle [latex]\,\beta \,[/latex] has the same cosine value as angle [latex]\,t;[/latex] the sine values are opposites.
Recall that an angle’s reference angle is the acute angle, [latex]\,t,[/latex] formed by the terminal side of the angle [latex]\,t\,[/latex] and the horizontal axis. A reference angle is always an angle between [latex]\,0\,[/latex] and [latex]\,90°,[/latex] or [latex]\,0\,[/latex] and [latex]\,\frac{\pi }{2}\,[/latex] radians. As we can see from (Figure), for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.
How To
Given an angle between [latex]\,0\,[/latex] and [latex]\,2\pi ,[/latex] find its reference angle.
- An angle in the first quadrant is its own reference angle.
- For an angle in the second or third quadrant, the reference angle is [latex]\,|\pi -t|\,[/latex] or [latex]\,|180°-t|.[/latex]
- For an angle in the fourth quadrant, the reference angle is [latex]\,2\pi -t\,[/latex] or [latex]\,360°-t.[/latex]
- If an angle is less than [latex]\,0\,[/latex] or greater than [latex]\,2\pi ,[/latex] add or subtract [latex]\,2\pi \,[/latex] as many times as needed to find an equivalent angle between [latex]\,0\,[/latex] and [latex]\,2\pi .[/latex]
Example 3 – Finding a Reference Angle
Find the reference angle of [latex]\,225°\,[/latex] as shown in (Figure).
Because [latex]\,225°\,[/latex] is in the third quadrant, the reference angle is
Try It
Find the reference angle of [latex]\,\frac{5\pi }{3}.[/latex]
Show answer
[latex]\frac{\pi }{3}[/latex]
7.4.3 – Using Reference Angles
Now let’s take a moment to reconsider the Ferris wheel introduced at the beginning of this section. Suppose a rider snaps a photograph while stopped twenty feet above ground level. The rider then rotates three-quarters of the way around the circle. What is the rider’s new elevation? To answer questions such as this one, we need to evaluate the sine or cosine functions at angles that are greater than 90 degrees or at a negative angle. Reference angles make it possible to evaluate trigonometric functions for angles outside the first quadrant. They can also be used to find [latex]\,\left(x,y\right)\,[/latex] coordinates for those angles. We will use the reference angle of the angle of rotation combined with the quadrant in which the terminal side of the angle lies.
Using Reference Angles to Evaluate Trigonometric Functions
We can find the cosine and sine of any angle in any quadrant if we know the cosine or sine of its reference angle. The absolute values of the cosine and sine of an angle are the same as those of the reference angle. The sign depends on the quadrant of the original angle. The cosine will be positive or negative depending on the sign of the x-values in that quadrant. The sine will be positive or negative depending on the sign of the y-values in that quadrant.
Using Reference Angles to Find Cosine and Sine
Angles have cosines and sines with the same absolute value as their reference angles. The sign (positive or negative) can be determined from the quadrant of the angle.
How To
Given an angle in standard position, find the reference angle, and the cosine and sine of the original angle.
- Measure the angle between the terminal side of the given angle and the horizontal axis. That is the reference angle.
- Determine the values of the cosine and sine of the reference angle.
- Give the cosine the same sign as the x-values in the quadrant of the original angle.
- Give the sine the same sign as the y-values in the quadrant of the original angle.
Example 4 – Using Reference Angles to Find Sine and Cosine
- Using a reference angle, find the exact value of [latex]\,\mathrm{cos}\left(150°\right)\,[/latex] and [latex]\,\text{sin}\left(150°\right).[/latex]
- Using the reference angle, find [latex]\,\mathrm{cos}\,\frac{5\pi }{4}\,[/latex] and [latex]\,\mathrm{sin}\,\frac{5\pi }{4}.[/latex]
- [latex]150°\,[/latex] is located in the second quadrant. The angle it makes with the x-axis is [latex]\,180°-150°=30°,[/latex] so the reference angle is [latex]\,30°.[/latex]
This tells us that [latex]\,150°\,[/latex] has the same sine and cosine values as [latex]\,30°,[/latex] except for the sign.
[latex]$$\begin{array}{ccc}\mathrm{cos}\left(30°\right)=\frac{\sqrt{3}}{2}& \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}& \mathrm{sin}\left(30°\right)=\frac{1}{2}\end{array}$$[/latex]Since [latex]\,150°\,[/latex] is in the second quadrant, the x-coordinate of the point on the circle is negative, so the cosine value is negative. The y-coordinate is positive, so the sine value is positive.
[latex]$$\begin{array}{ccc}\mathrm{cos}\left(150°\right)=-\frac{\sqrt{3}}{2}& \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}& \mathrm{sin}\left(150°\right)=\frac{1}{2}\end{array}$$[/latex] - [latex]\frac{5\pi }{4}\,[/latex] is in the third quadrant. Its reference angle is [latex]\,\frac{5\pi }{4}-\pi =\frac{\pi }{4}.\,[/latex] The cosine and sine of [latex]\,\frac{\pi }{4}\,[/latex] are both [latex]\,\frac{\sqrt{2}}{2}.\,[/latex] In the third quadrant, both [latex]\,x\,[/latex] and [latex]\,y\,[/latex] are negative, so:
[latex]$$\begin{array}{ccc}\mathrm{cos}\phantom{\rule{0.03em}{0ex}}\frac{5\pi }{4}=-\frac{\sqrt{2}}{2}& \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}& \mathrm{sin}\phantom{\rule{0.03em}{0ex}}\frac{5\pi }{4}=-\frac{\sqrt{2}}{2}\end{array}$$[/latex]
Try It
- Use the reference angle of [latex]\,315°\,[/latex] to find [latex]\,\mathrm{cos}\left(315°\right)\,[/latex] and [latex]\,\mathrm{sin}\left(315°\right).[/latex]
- Use the reference angle of [latex]\,-\frac{\pi }{6}\,[/latex] to find [latex]\,\mathrm{cos}\left(-\frac{\pi }{6}\right)\,[/latex] and [latex]\,\mathrm{sin}\left(-\frac{\pi }{6}\right).[/latex]
Show answer
- [latex]\text{cos}\left(315°\right)=\frac{\sqrt{2}}{2}, \text{sin}\left(315°\right)=\frac{–\sqrt{2}}{2}[/latex]
- [latex]\text{cos}\left(-\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2},\mathrm{sin}\left(-\frac{\pi }{6}\right)=-\frac{1}{2}[/latex]
Using Reference Angles to Find Coordinates
Now that we have learned how to find the cosine and sine values for special angles in the first quadrant, we can use symmetry and reference angles to fill in cosine and sine values for the rest of the special angles on the unit circle. They are shown in (Figure). Take time to learn the [latex]\,\left(x,y\right)\,[/latex] coordinates of all of the major angles in the first quadrant.
Key Equations
| Cosine | [latex]\mathrm{cos}\,t=x[/latex] |
| Sine | [latex]\mathrm{sin}\,t=y[/latex] |
| Pythagorean Identity | [latex]{\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t=1[/latex] |
Key Concepts
- Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit.
- Using the unit circle, 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] whereas the cosine of an angle [latex]\,t\,[/latex] equals the x-value of the endpoint. See (Example 1).
- The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. See (Example 2).
- The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.
- The signs of the sine and cosine are determined from the x– and y-values in the quadrant of the original angle.
- An angle’s reference angle is the size angle, [latex]\,t,[/latex] formed by the terminal side of the angle [latex]\,t\,[/latex] and the horizontal axis. See (Example 3).
- Reference angles can be used to find the sine and cosine of the original angle. See (Example 4).
- Reference angles can also be used to find the coordinates of a point on a circle. See (Example 5).
Glossary
- cosine function
- the x-value of the point on a unit circle corresponding to a given angle
- Pythagorean Identity
- a corollary of the Pythagorean Theorem stating that the square of the cosine of a given angle plus the square of the sine of that angle equals 1
- sine function
- the y-value of the point on a unit circle corresponding to a given angle