1. What is the Unit Circle?

The unit circle is simply a circle with a radius of 1 unit. There is a convenient function that defines all of the (x,y) pairs that lie on the circle: \[ x^2 + y^2 = 1\] This could be rearranged to be written as a non-implicit function: \[ y = \pm\sqrt{1 - x^2}\] Plotting this pair of functions (\(\pm\)) gives the unit circle:

Any pair of x and y that satisfy this function map a position that is exactly 1 unit from the origin (it is on the unit circle).

2. Can you define a position on the circle with just one value?

For these points, it takes two pieces of information (x and y coordinates) to define their location, but for now we are only interested in points on the unit circle. This allows us to define a unique position with the single value of its angle \(\theta\) from the positive x-axis (this is just a common standard from which to measure the angle \(\theta\)).

3. What if you are given the angle \(\theta\) and want to calculate the vertical and horizontal position?

This is exactly what the sine and cosine functions are defined to do! Sine and Cosine are the fundamental trigonometric (trig) functions and are definded to calculate the vertical and horizontal distances to a position described by an angle.

The cosine function is defined to output the horizontal distance to the point on the unit circle described by the angle \(\theta\).

The sine function is defined to output the vertical distance to the point on the unit circle described by the angle \(\theta\).

\( x = \cos(\theta)\)
\( y = \sin(\theta)\)

\( ( x , y ) = (\) \(\cos(\theta)\) , \(\sin(\theta)\) \()\)

Because \(x = cos(\theta)\) and \(y = sin(\theta)\) these can be substituted into \( x^2 + y^2 = 1\) to get the following relationship: \[ \cos(\theta)^2 + \sin(\theta)^2 = 1 \] which is useful for manipulating equations invololving trigonometric functions.

4. The trig functions do not create unique outputs for every angle.

If you take the horizontal component for example, \(cos(θ)\) an output of 0.5 at both +60° and -60°.

Because you can go 360° in either direction and end up at the same place, the trig functions give the same output to an angle \(\pm\) any multiple of 360°.

5. What is the motion of the Vertical component while the angle is changing?

It is often important to understand how one or the other component of position varies with the angle \(\theta\); for example the Scotch Yoke. We can get a visualization of the vertical motion by tracing it while changing the angle \(\theta\).

This is called sinusiodal motion or simple harmonic motion. It repeats every 360° and oscillates between \(\pm\) 1.

For a great explanation of how any sinusoidal motion can be described by this rotating wheel model, see this animation by Bartosz Ciechanowski. It shows how the amplitude of motion can by determined by the radius of the wheel and the frequency by the speed of rotation.

6. How do the Vertical and Horizontal motions compare?

The horizontal motion (described by \(\cos(\theta)\) ) is also sinusoidal; however, it is shifted from the vertical motion by 90°.

Whenever \(\sin(\theta)\) is at zero \(\cos(\theta)\) is at a maxima (±1). Whenever \(\sin(\theta)\) is at a maxima (±1), \(\cos(\theta)\) is at 0.

The property of sine and cosine being related by a shift of 90° can be written as relations which can be useful for converting between them. \[ \sin(\theta) = cos(90^\circ - \theta) \] \[ \cos(\theta) = sin(90^\circ - \theta) \]

7. What if you want to know the Slope to a position?

The tangent function is defined as the ratio of \(\frac{Sin(\theta)}{Cos(\theta)}\) which means it describes the slope ( \(\frac{rise}{run}\) ) of the line from the origin to the point defined by the angle \(\theta\). \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]

8. What does the tangent graph look like?

When a line is horizontal, its slope - and \(\tan(\theta)\) - is zero. When a line is vertical, the slope is undefined because it could be either positive or negative infinity. If we are rotating in the positive θ direction from the x-axis, then it is going to approach infinity as θ approaches 90°. If we approach 90° from the negative x-axis with a decreasing θ, then the slope will approach negative infinity.

The graph becomes undefined at θ = 90°, 270°, 450°, etc, and is be zero at θ = multiples of 180°.

9. Why is it called Tangent?

If you look at a vertical line tangent to the unit circle, any point on it can be described by \(\tan(\theta)\) where θ is the angle made when pointing from the origin to the point.