Note that this definition allows us to use angles which are not acute or in other words, angles in standard position whose terminal side is not in the first quadrant.

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 3, 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]

Figure 3:  Illustration of an angle t, with terminal side length equal to 1, and an arc created by angle with length t.

Important Note:  Because it is understood that sine and cosine are functions, we do not always need to write them with parentheses: [latex]\mathrm{sin}\text{ }t[/latex] is the same as [latex]\mathrm{sin}\left(t\right)[/latex] and [latex]\mathrm{cos}\text{ }t[/latex] is the same as [latex]\mathrm{cos}\left(t\right).[/latex] Likewise, [latex]{\mathrm{cos}}^{2}\left(t\right)[/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.

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 4. Find [latex]\mathrm{cos}\left(t\right)[/latex] and [latex]\mathrm{sin}\left(t\right).[/latex]

Figure 4:  Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the given point

Show Solution

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

[latex]\begin{align*}x&=\mathrm{cos}\left(t\right)=\frac{1}{2}\\ y&=\mathrm{sin}\left(t\right)=\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{2} \end{align*}[/latex]

Try It #1

A certain angle [latex]t[/latex] corresponds to a point on the unit circle at [latex]\left(-\frac{\sqrt[\leftroot{1}\uproot{2} ]{2}}{2},\frac{\sqrt[\leftroot{1}\uproot{2} ]{2}}{2}\right)[/latex] as shown in Figure 5. Find [latex]\mathrm{cos}\left(t\right)[/latex] and [latex]\mathrm{sin}\left(t\right).[/latex]

Figure 5:  Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the given point.

Show Solution

[latex]\mathrm{cos}\left(t\right)=-\frac{\sqrt[\leftroot{1}\uproot{2} ]{2}}{2},\text{ }\mathrm{sin}\left(t\right)=\frac{\sqrt[\leftroot{1}\uproot{2} ]{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]\mathrm{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), as shown in Figure 6.

Figure 6:  Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (0, 1).

Using our definitions of cosine and sine,

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

The cosine of 90° is 0; the sine of 90° 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}\left(t\right)[/latex] and [latex]y=\mathrm{sin}\left(t\right),[/latex] we can substitute for [latex]x[/latex] and [latex]y[/latex] to get [latex]\mathrm{cos}^{2}\left(t\right)+\mathrm{sin}^{2}\left(t\right)=1.[/latex]  This equation, [latex]\mathrm{cos}^{2}\left(t\right)+\mathrm{sin}^{2}\left(t\right)=1,[/latex] is known as the Pythagorean Identity. See Figure 7.

Figure 7

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.

Example 3:  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 8.

Figure 8:  Graph of a unit circle with an angle that intersects the circle at a point with the y-coordinate equal to 3/7.

Substituting the known value for sine into the Pythagorean Identity,

[latex]\begin{align*}\mathrm{cos}^{2}\left(t\right)+\mathrm{sin}^{2}\left(t\right)&=1\\ \mathrm{cos}^{2}\left(t\right)+\frac{9}{49}&=1\\\mathrm{cos}^{2}\left(t\right)&=\frac{40}{49}\\\mathrm{cos}\left(t\right)&=±\sqrt[\leftroot{1}\uproot{2} ]{\frac{40}{49}}=±\frac{\sqrt[\leftroot{1}\uproot{2} ]{40}}{7}=±\frac{2\text{ }\sqrt[\leftroot{1}\uproot{2} ]{10}}{7}\end{align*}[/latex][latex]\\[/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.  So [latex] \mathrm{cos}\left(t\right)=-\frac{2\text{ }\sqrt[\leftroot{1}\uproot{2} ]{10}}{7}.[/latex]

Finding Sines and Cosines of 45° Angles and 30° and 60° Angles

We have already found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of the unit circle. Figure 9 summarizes these values.

Figure 9:  Graph of a quarter circle with angles of 0, 30, 45, 60, and 90 degrees inscribed. Equivalence of angles in radians shown.

Using a Calculator to Find Sine and Cosine

To find the cosine and sine of angles other than the special angles, we turn to a computer or calculator. Be aware: Most calculators can be set into “degree” or “radian” mode, which tells the calculator the units for the input value. When we evaluate [latex]\mathrm{cos}\left(30\right)[/latex] on our calculator, it will evaluate it as the cosine of 30 degrees if the calculator is in degree mode, or the cosine of 30 radians if the calculator is in radian mode.

Try It #4

Evaluate [latex]\mathrm{sin}\left(2.3\right).[/latex]

Show Solution

approximately 0.74571

Reference Angles

For any given angle in the first quadrant, there is an angle in the second quadrant with the same y-value and therefore the same sine value. Because the sine value is the y-coordinate on the unit circle, the other angle with the same sine will share the same y-value, but have the opposite x-value. Therefore, its cosine value will be the opposite of the first angle’s cosine value.

Likewise, there will be an angle in the fourth quadrant with the same x-value and therefore the same cosine as the original angle in the first quadrant. 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 11, 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.

[latex]\begin{array}{lll}\mathrm{sin}\left(t\right)=\text{ }\mathrm{sin}\left(\alpha \right)\text{ }\hfill & \text{and}\hfill & \mathrm{cos}\left(t\right)=-\mathrm{cos}\left(\alpha \right)\hfill \\ \mathrm{sin}\left(t\right)=-\mathrm{sin}\left(\beta \right)\hfill & \text{and}\hfill & \mathrm{cos}\left(t\right)=\text{ }\mathrm{cos}\left(\beta \right)\hfill \end{array}[/latex]

Figure 11

Finding Reference Angles

An angle’s reference angle is the measure of the smallest, positive, acute angle [latex]t[/latex] formed by the terminal side of the angle [latex]t[/latex] and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants.

Definition

An angle’s reference angle is the size of the smallest acute angle, [latex]{t}^{\prime },[/latex] formed by the terminal side of the angle [latex]t[/latex] and the horizontal axis.

We can see that reference angle is always an angle between [latex]0^{\circ}[/latex] and [latex]90^{\circ},[/latex] or [latex]0[/latex] and [latex]\frac{\pi }{2}[/latex] radians. As we can see from Figure 12, for any angle in quadrants II, III, or IV, there is a reference angle in quadrant I.

Figure 12:  Reference Angles

How To

Given an angle between [latex]0[/latex] and [latex]2\pi,[/latex] or between [latex]0^{\circ}[/latex] and [latex]360^{\circ},[/latex] find its reference angle.

1. An angle in the first quadrant is its own reference angle.
2. For an angle in the second quadrant, the reference angle is [latex]\pi-t [/latex] or [latex]180^{\circ}-t.[/latex]
3. For an angle in the third quadrant, the reference angle is [latex]t-\pi[/latex] or [latex]t-180^{\circ}.[/latex]
4. For an angle in the fourth quadrant, the reference angle is [latex]2\pi -t[/latex] or [latex]360^{\circ}-t.[/latex]

Given an angle less than [latex]0[/latex] or greater than [latex]2\pi,[/latex] or less than [latex]0^{\circ}[/latex] or greater than [latex]360^{\circ},[/latex] find its reference angle.

1. 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 a coterminal angle between [latex]0[/latex] and [latex]2\pi .[/latex]  Similarly, if working in degrees add or subtract [latex]360^{\circ}[/latex] as many times as needed to find a coterminal angle between [latex]0^{\circ}[/latex] and [latex]360^{\circ}.[/latex]
2. Once you have an angle between [latex]0[/latex] and [latex]2\pi,[/latex] or between [latex]0^{\circ}[/latex] and [latex]360^{\circ},[/latex] follow steps 1 – 4 above.

Example 5:  Finding a Reference Angle

Find the reference angle of [latex]225^{\circ}[/latex] as shown in Figure 13.

Figure 13:  Graph of circle with 225 degree angle inscribed.

Show Solution

Because [latex]225^{\circ}[/latex] is in the third quadrant, the reference angle is [latex] 225^{\circ} -180^{\circ}=45^{\circ}.[/latex]

Try It # 5

Find the reference angle of [latex]\frac{5\pi }{3}.[/latex]

Show Solution

[latex]\frac{\pi }{3}[/latex]

Using Reference Angles to Find Coordinates

Now that we have learned how to find the cosine and sine values for angles whose reference angles are acute special angles, the rest of the special angles on the unit circle can be determine. They are shown in Figure 14. Take time to learn the [latex]\left(x,y\right)[/latex] coordinates of all of the major angles in the first quadrant.

Figure 14: Special angles and coordinates of corresponding points on the unit circle

In addition to learning the values for special angles, we can use reference angles to find [latex]\left(x,y\right)[/latex] coordinates of any point on the unit circle, using what we know of reference angles along with the identities

[latex]\begin{align*}x&=\mathrm{cos}\left(t\right) \\ y&=\mathrm{sin}\left(t\right).\end{align*}[/latex][latex]\\[/latex]

First we find the reference angle corresponding to the given angle. Then we take the sine and cosine values of the reference angle, and give them the signs corresponding to the y– and x-values of the quadrant.

How To

Given the angle of a point on a unit circle, find the [latex]\left(x,y\right)[/latex] coordinates of the point.

1. Find the reference angle by measuring the smallest angle to the x-axis.
2. Find the cosine and sine of the reference angle.
3. Determine the appropriate signs for [latex]x[/latex] and [latex]y[/latex] in the given quadrant.

Example 8:  Using the Unit Circle to Find Coordinates

Find the coordinates of the point on the unit circle at an angle of [latex]\frac{7\pi }{6}.[/latex]

Show Solution

We know that the angle [latex]\frac{7\pi }{6}[/latex] is in the third quadrant.

First, let’s find the reference angle by measuring the angle to the x-axis. To find the reference angle of an angle whose terminal side is in quadrant III, we find the difference of the angle and [latex]\pi .[/latex]

[latex]\frac{7\pi }{6}-\pi =\frac{\pi }{6}[/latex][latex]\\[/latex]

Next, we will find the cosine and sine of the reference angle:

[latex]\mathrm{cos}\left(\frac{\pi }{6}\right)=\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{2}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\mathrm{sin}\left(\frac{\pi }{6}\right)=\frac{1}{2}[/latex][latex]\\[/latex]

We must determine the appropriate signs for x and y in the given quadrant. Because our original angle is in the third quadrant, where both [latex]x[/latex] and [latex]y[/latex] are negative, both cosine and sine are negative.

[latex]\begin{align*}\mathrm{cos}\left(\frac{7\pi }{6}\right)&=-\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{2} \\ \mathrm{sin}\left(\frac{7\pi }{6}\right)&=-\frac{1}{2}\end{align*}[/latex][latex]\\[/latex]

Now we can calculate the [latex]\left(x,y\right)[/latex] coordinates using the identities [latex]x=\mathrm{cos}\left(\theta\right) [/latex] and [latex]y=\mathrm{sin}\left(\theta\right) .[/latex]

The coordinates of the point are [latex]\left(-\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{2},-\frac{1}{2}\right)[/latex] on the unit circle.

Try It # 8

Find the coordinates of the point on the unit circle at an angle of [latex]\frac{5\pi }{3}.[/latex]

Show Solution

[latex]\left(\frac{1}{2},-\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{2}\right)[/latex]

Circles with Radius Different Than 1

Suppose we have a circle centered at the origin other than the unit circle. How can we find the coordinates of a point where the terminal side of the angle in standard position intersects the circle? We know from Section 3.1 that the sine and cosine values of an acute angle do not change regardless of the size of the triangle. This means that [latex]\mathrm{cos}\left(t\right)[/latex] will be the same whether [latex]t[/latex] is in a triangle with small or large sides. However, it is clear that the coordinates of the point of intersection of the terminal side with the circle of radius r will not be the same as the coordinates of the point on the unit circle.

We can apply similar reasoning to considering values of sine and cosine values on a circle that is not a unit circle as we did earlier in this section. By drawing an angle in standard position in quadrant 1, we can drop a perpendicular from the point of intersection of the terminal side with the circle whose center is at the origin to the x axis. If the point on the circle is (x, y), then the sides of the right triangle formed by dropping the perpendicular will also be [latex]x\text{ and }y.[/latex] Now, instead of the radius being 1, we would have a radius of [latex]r.[/latex] Therefore, [latex]\mathrm{cos}\left(t\right)=\frac{x}{r}\text{ and }\mathrm{sin}\left(t\right)=\frac{y}{r}.[/latex] This means that [latex]x=r\mathrm{cos}\left(t\right)\text{ and }y=r\mathrm{sin}\left(t\right).[/latex]

Definition

A point [latex]\left(x,y\right)[/latex] on the circle, centered at the origin, with radius [latex]r[/latex] corresponds to

[latex]\begin{align*}x&=r\mathrm{cos}\left(t\right)\text{ and }\\y&=r\mathrm{sin}\left(t\right)\end{align*}[/latex][latex]\\[/latex]

where [latex]t[/latex] is the angle in standard position.

Example 9:  Finding Coordinates on a Circle with Radius [latex]r[/latex]

Find the coordinates of the point on the circle with radius 5 at an angle of [latex]\frac{7\pi}{6}.[/latex]

Show Solution

We already know from Example 7 that:

[latex]\begin{align*}\mathrm{cos}\left(\frac{7\pi }{6}\right)&=-\frac{\sqrt[\leftroot{1}\uproot{2} ]{3}}{2} \\ \mathrm{sin}\left(\frac{7\pi }{6}\right)&=-\frac{1}{2}\end{align*}[/latex][latex]\\[/latex]

Now we can calculate the [latex]\left(x,y\right)[/latex] coordinates using the identities

[latex]x=r\mathrm{cos}\left(\theta\right) [/latex] and [latex]y=r\mathrm{sin}\left(\theta\right) .[/latex]

The coordinates of the point are [latex]\left(-\frac{5\text{ }\sqrt[\leftroot{1}\uproot{2} ]{3}}{2},-\frac{5}{2}\right)[/latex] on the circle with radius 5.

Try It # 9

Find the coordinates of the point on the circle with radius 7 at an angle of [latex]\frac{5\pi }{3}.[/latex]

Show Solution

[latex]\left(\frac{7}{2},-\frac{7\text{ }\sqrt[\leftroot{1}\uproot{2} ]{3}}{2}\right)[/latex]

Even and Odd Functions

To be able to use our sine and cosine functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.

Recall that:

• An even function is one in which [latex]f\left(-x\right)=f\left(x\right).[/latex]
• An odd function is one in which [latex]f\left(-x\right)=-f\left(x\right).[/latex]

We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in  Figure 15. The sine of the positive angle is [latex]y.[/latex] The sine of the negative angle is [latex]-y.[/latex]The sine function, then, is an odd function.  The cosine of the positive angle is [latex]x[/latex], as is the cosine of the negative angle.  Therefore, the cosine function is an even function.

Figure 15

We can summarize this by saying:

[latex]\begin{align*}\mathrm{sin}\left(-x\right)&=-\mathrm{sin}\left(x\right)\\ \mathrm{cos}\left(-x\right)&=\mathrm{cos}\left(x\right)\end{align*}[/latex]

Access these online resources for additional instruction and practice with sine and cosine functions.

• Trigonometric Functions Using the Unit Circle
• Sine and Cosine from the Unit Circle
• Sine and Cosine from the Unit Circle and Multiples of Pi Divided by Six
• Sine and Cosine from the Unit Circle and Multiples of Pi Divided by Four
• Trigonometric Functions Using Reference Angles

Key Equations

Cosine	[latex]\mathrm{cos}\left(t\right)=x[/latex]
Sine	[latex]\mathrm{sin}\left(t\right)=y[/latex]
Pythagorean Identity	[latex]{\mathrm{cos}}^{2}\left(t\right)+{\mathrm{sin}}^{2}\left(t\right)=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.
• When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles.
• Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known.
• The domain of the sine and cosine functions is all real numbers.
• The range of both the sine and cosine functions is [latex]\left[-1,1\right].[/latex]
• 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.
• Reference angles can be used to find the sine and cosine of the original angle.
• Reference angles can also be used to find the coordinates of a point on a unit circle.
• When the radius of a circle centered at the origin is not 1, we can find coordinates of a point on the circle by multiplying the sine and cosine of the angle by [latex]r.[/latex]

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

unit circle

a circle with a center at [latex]\left(0,0\right)[/latex] and radius 1.