Trigonometric Functions: Unit Circle Approach

The Unit Circle is a circle with its center at the origin (0,0) and a radius of one unit.

Unit Circle

Angles are always measured from the positive x-axis (also called the "right horizon"). Angles measured counterclockwise have positive values; angles measured clockwise have negative values.

Defining sine and cosine in terms of the unit circle

In the unit circle shown here, a unit-length radius has been drawn from the origin to a point (x, y) on the circle.

Defining sine and cosine

A line perpendicular to the x-axis, drawn through the point (x, y), intersects the x-axis at the point with the abscissa x. Similarly, a line perpendicular to the y-axis intersects the y-axis at the point with the ordinate y. The angle between the x-axis and the radius is $\alpha$.

So, we can say that the sine of an angle is the ordinate of the point on the unit circle at that angle, and the cosine of an angle is the abscissa of the point on the unit circle at that angle.

We define the basic trigonometric functions of any angle $\alpha$ as follows:

Template:TrigBoxOpen ${\begin{matrix}\mathrm {sine:} &\sin(\alpha )&=&y\\\mathrm {cosine:} &\cos(\alpha )&=&x\\\end{matrix}}$

$\tan(\theta )$ can be algebraically defined.

$\tan(\theta )={\frac {\sin(\theta )}{\cos(\theta )}}\qquad \cos(\theta )\neq 0$

$\tan(\alpha )={\frac {y}{x}}\qquad x\neq 0$

These three trigonometric functions can be used whether the angle is measured in degrees or radians as long as it specified which, when calculating trigonometric functions from angles or vice versa.

Alternative definitions

• A previous chapter used Soh-Cah-Toa to define the trigonometric functions. The advantage of the unit circle is that θ can be extended outside the first quadrant $\left[0,{\frac {\pi }{2}}\right]$, which allows us to define these functions on the interval $(-\infty ,\infty )$.
• If trigonometry is applied to vectors, it is more convenient, if the radius of the circle is not equal to unity. For example, if vector A has magnitude $A=\left|\mathbf {A} \right\vert$:

Template:TrigBoxOpen ${\begin{matrix}x=r\cos(\alpha )&\to &\mathbf {A} _{x}=\mathbf {A} \cos(\alpha )\\y=r\sin(\alpha )&\to &\mathbf {A} _{y}=\mathbf {A} \sin(\alpha )\\r={\sqrt {x^{2}+y^{2}}}&\to &\mathbf {A} ={\sqrt {\left(\mathbf {A} _{x}\right)^{2}+\left(\mathbf {A} _{y}\right)^{2}}}\\\end{matrix}}$ Template:TrigBoxClose

It is important to know why the above equations are true. Knowing $\cos(\alpha )={\frac {x}{r}}$, $r\cdot \cos(\alpha )=r\cdot {\frac {x}{r}}=x$. The same could be said for the definition for $y$. Finally, the final line is the Pythagorean identity.

Video links

More about this topic can be found at the 'Khan Academy:'

• The unit circle definition of trigonometric function
• Unit circle definition of trig functions

Some values for sine and cosine

A unit circle with certain exact values marked on it is below:

Labeled unit circle

Unit circles form the basis of most analog clocks and animations on computers since the cos and sin correspond to the x and y positions of the end of the line segments representing the hands of the clock.

The unit circle on the left has the degree, the radian, and the coordinate value on the unit circle. For a coordinate value $(x,y)$, if walking around the circle ${\frac {\pi }{3}}$ radians anti-clockwise from the horizontal axis, the coordinate value at which the person walked around the circle is $\left({\frac {1}{2}},{\frac {\sqrt {3}}{2}}\right)$.

Remember that on a unit circle, the angle $\theta$ anti-clockwise from the horizontal axis gives $\cos(\theta )=x$ and $\sin(\theta )=y$. The same is true for radians. As such, for $(x,y)$ corresponding to $(\cos(\theta ),\sin(\theta ))$ on the unit circle, ${\frac {\pi }{3}}$ radians when substituted in the cosine or sine function is the coordinate value on the unit circle. That is:

$\left(\cos \left({\frac {\pi }{3}}\right),\sin \left({\frac {\pi }{3}}\right)\right)\to \left({\frac {1}{2}},{\frac {\sqrt {3}}{2}}\right)$, OR, equivalently,

$\left(\cos \left(60^{\circ }\right),\sin \left(60^{\circ }\right)\right)\to \left({\frac {1}{2}},{\frac {\sqrt {3}}{2}}\right)$

The unit circle is very useful to your mathematical studies of trigonometry because it tells you the EXACT value of certain angles. Later, you will learn how to find other ratios of angles and radians without needing to rely on the special values of the unit circle – 30, 45, 60, and 90.

It is worth your time memorizing some of the values of sine and cosine on the unit circle (cosine is equal to $x$ while sine is equal to $y$). You should at least become familiar with the values for $0^{\circ },30^{\circ }45^{\circ },90^{\circ }$ and know where ${\frac {\pi }{2}},\pi ,{\frac {3\pi }{2}},2\pi$ are on the unit circle.

If you have some trouble memorizing the values, here are some helpful hints and patterns. Try to find some more other than what is listed.

• The coordinate value $(x,y)$ on the unit circle has the ${\text{numerator}}$ decrease from ${\sqrt {3}}$ in the $x$-value coordinate at ${\frac {\pi }{6}}$ to ${\sqrt {1}}=1$ at ${\frac {\pi }{3}}$. The denominator is always $2$. Here, see what we mean. Ignore the y-value for now:
  • $\left({\frac {\sqrt {3}}{2}},\sin \left({\frac {\pi }{6}}\right)\right)$, $\left({\frac {\sqrt {2}}{2}},\sin \left({\frac {\pi }{4}}\right)\right)$, $\left({\frac {1}{2}},\sin \left({\frac {\pi }{3}}\right)\right)$.
• Like the $x$-value coordinate, the $y$-value coordinate has a pattern in the numerator. As the angle increases, between $30^{\circ }$ and $60^{\circ }$ (inclusive), the ${\text{numerator}}$ increases from ${\sqrt {1}}=1$ in the $y$-value coordinate at ${\frac {\pi }{6}}$ to ${\sqrt {3}}$ at ${\frac {\pi }{3}}$. From the above bullet, putting it together, you get the following pattern:
  • $\left({\frac {\sqrt {3}}{2}},{\frac {1}{2}}\right)$, $\left({\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}\right)$, $\left({\frac {1}{2}},{\frac {\sqrt {3}}{2}}\right)$.
• Certain radian values are simply reflections of the other. Take a look at the ones with the same denominator and see if you can correspond any patterns to what you see.

If you are to ever be tested on this, make a quick sketch of the first quadrant of the circle, and remember the pattern that underlies the unit circle. Template:Clear

Clock Hands:

• What angle (a) in degrees and (b) in radians do the hour and minute hands of a clock move through in:
• One hour
• One minute
• How far apart in degrees are the hour and minute hand at five minutes past five? (you should take into account that the hour hand is not exactly at the five, but has moved a little further)

Resources

• Trigonometric Functions: Unit Circle Approach. Written notes created by Professor Esparza, UTSA.