Unit Circle

The Unit Circle is a a circle with its center at the origin \((0,0)\) and a radius of one unit. It is used to determine the angle values for all trigonometric ratios.

We can use the point on the Unit Circle to identify the cos and sin values; \(\cos\theta\) is equal to the \(x\)-coordinate (or base) while \(\sin\theta\) is equal to the \(y\)-coordinate (or altitude).

Characteristics of a Unit Circle

Terminal Arm

The ray of an angle attached to the origin that travels around the circle

Initial Arm

The ray of an angle that has a default position of \(0°\) at the positive \(x\)-axis

Angle Rotation

As angles are always measured from the positive \(x\)-axis, angles measured counterclockwise have positive values while those measured clockwise have negative values.

Positive Angle Rotation occurs when the terminal arm moves counterclockwise
Negative Angle Rotation occurs when the terminal arm moves clockwise

CAST

An acronym used to memorize which trigonometric ratios yield positive results in which quadrant(s):

C: Quadrant \(4\), will yield positive results for \(\cos\) and negative results for \(\sin\) and \(\tan\)
A: Quadrant \(1\), will yield positive results for all trigonometric functions
S: Quadrant \(2\), will yield positive results for \(\sin\) and negative results for \(\cos\) and \(\tan\)
T: Quadrant \(3\), will yield positive results for \(\tan\) and negative results for \(\cos\) and \(\sin\)

Standard Position

The angle's position where the vertex is located at the origin, one of the rays lies on the positive \(x\)-axis, and the other ray is formed by travelling counter-clockwise around the circle
All angles must be between \(0°\) and \(360°\)

Principle Angle (\(\theta\))

The angle of a circle that lies between the initial arm and the terminal arm in standard position
As the terminal arm moves in a counter-clockwise position, this angle must always be positive!!
Can be represented algebraically as \(0°≤\theta≤360°\)

Related Acute Angle (\(\beta\))

Also referred to as the reference angle, this is the acute angle made between the \(x\)-axis and the terminal arm when the principle angle is greater than \(90°\)
The formula used to determine the reference angle depends on the position of the principle angle:

Quadrant 2: \(\beta = 180° - \theta\)
Quadrant 3: \(\beta = \theta - 180°\)
Quadrant 4: \(\beta = 360° - \theta\)

The reference angle must ALWAYS be positive!!

Reflex Angles

Angles greater than \(180°\)

Co-Terminal Angles

Angles that can be found be adding or subtracting multiples of \(360°\) to the principle angle

Diagram of Unit Circle with key features including terminal arm, principal angle, and co-terminal angle.

For a principal angle of \(110^{\circ}\):

Determine the reference angle and 2 co-terminal angles
Sketch a graph displaying both the terminal and reference angles

Show Answer

i. Since the principle angle can be found in the second quadrant, we can determine the reference angle by subtracting the principle angle from \(180°\):

\(β = 180° - 110°\)

\(β = 70°\)

We can determine the first co-terminal angle by adding \(360°\) to the principle angle:

\(\text{CA}_1 = 110° + 360°\)

\(\text{CA}_1 = 470°\)

We can then determine the second co-terminal angle by subtracting \(360°\) from the principle angle:

\(\text{CA}_2 = 110° - 360°\)

\(\text{CA}_2 = -250°\)

Therefore, we can determine the reference angle as \(\boldsymbol{70°}\) and the co-terminal angles as \(\boldsymbol{470°}\) and \(\boldsymbol{-250°}\).

ii. Using these angles, we can sketch the graph as such:

Unit Circle showing terminal angle of 70° and corresponding reference angle.

Trigonometric Ratios

As stated above, \(\cos\theta\) and \(\sin\theta\) are respectively used to determine the \(x\) and \(y\)-coordinates of a point on the unit circle. In addition, \(\tan \theta\) is used to determine the slope of the terminal arm. Trigonemtric ratios can be used to describe the relationship between these variables:

\(\sin \theta = \cfrac{\textcolor{red}{\text{opposite}}}{\textcolor{blue}{\text{hypotenuse}}} = \cfrac{\textcolor{red}{y}}{\textcolor{blue}{r}}\)

\(\cos \theta = \cfrac{\textcolor{green}{\text{adjacent}}}{\textcolor{blue}{\text{hypotenuse}}} = \cfrac{\textcolor{green}{x}}{\textcolor{blue}{r}}\)

\(\tan \theta = \cfrac{\textcolor{red}{\text{opposite}}}{\textcolor{green}{\text{adjacent}}} = \cfrac{\textcolor{red}{y}}{\textcolor{green}{x}}\)

A right triangle can be formed when given at least \(2\) pieces of information. In order to find the other piece of information, we can use the Pythagorean Theorem:

\(\textcolor{green}{x}^2 + \textcolor{red}{y}^2 = \textcolor{blue}{r}^2\)

Example

For the point \(P(-3,-5)\), find the \(\cos\), \(\sin\), and \(\tan\) ratios.

First, we need to identify the hypotenuse. To do so, we can use the Pythagorean Theorem:

\(\textcolor{green}{x}² + \textcolor{red}{y}² = \textcolor{blue}{r}²\)

\((\textcolor{green}{-3})² + (\textcolor{red}{-5})² = \textcolor{blue}{r}²\)

\(9 + 25 = \textcolor{blue}{r}²\)

\(\sqrt{\textcolor{blue}{r}²} = \sqrt{34}\)

\(\textcolor{blue}{r = \sqrt{34}}\)

Using the hypotenesue, we can identify all \(3\) trigonometric ratios:

\(\sin \theta = \cfrac{\textcolor{red}{-5}}{\textcolor{blue}{\sqrt{34}}}\)

\(\cos \theta = \cfrac{\textcolor{green}{-3}}{\textcolor{blue}{\sqrt{34}}}\)

\(\tan \theta = \cfrac{\textcolor{red}{-5}}{\textcolor{green}{-3}} = \cfrac{\textcolor{red}{5}}{\textcolor{green}{3}}\)

Therefore, we can determine the respective trigonometric ratios for \(\sin\), \(\cos\), and \(\tan\) are \(\boldsymbol{\cfrac{\textcolor{red}{-5}}{\textcolor{blue}{\sqrt{34}}}}\), \(\boldsymbol{\cfrac{\textcolor{green}{-3}}{\textcolor{blue}{\sqrt{34}}}}\), and \(\boldsymbol{\cfrac{\textcolor{red}{5}}{\textcolor{green}{3}}}\) .

For the ratio \(\sin \theta = -\cfrac{2}{5}\), the angle, \(\theta\) is in standard position.

Identify how many answers for \(\theta\) there can be.
Find all possible measures of \(\theta\) in the given domain.
Are these measure(s) acute, obtuse, or reflex?
Sketch a graph with these angles.

Show Answer

i. There are \(2\) answers for \(\theta\). We can determine this by looking at the ratio, \(\sin \theta = \cfrac{\textcolor{red}{y}}{\textcolor{blue}{r}}\).

Since \(r\) is always going to be positive, this means that \(y\) is always going to be negative. As a result, we can determine Quadrants \(3\) and \(4\) fulfill both of these conditions.

ii. In order to find the first measure, we need to find the inverse of the ratio:

\(\theta_1 = \sin^{-1}\left(-\cfrac{\textcolor{red}{y}}{\textcolor{blue}{r}}\right)\)

\(\theta_1 = \sin^{-1}\left(-\cfrac{\textcolor{red}{2}}{\textcolor{blue}{5}}\right)\)

\(\theta_1 = -24°\)

Since all angles in Standard Position must be betwen \(0°\) and \(360°\), we can find the positive equivalent in Quadrant \(4\) by adding \(360°\) to \(\theta_1\):

\(\theta_1 = -24° + 360°\)

\(\theta_1 = 336°\)

In order to get the corresponding angle in Quadrant \(3\), we can add \(180°\) to \(\theta_1\):

\(\theta_2 = 180° + 24°\)

\(\theta_2 = 204°\)

Therefore, we can determine that the \(2\) measures for \(\theta\) are \(\boldsymbol{336°}\) and \(\boldsymbol{204°}\).

iii. We can determine that both of these measures are reflex angles since they are greater than \(180°\).

iv. Using the information determined above, we can sketch the graph:

Unit Circle showing equivalent angles of 336° and 204°.

Using this diagram, we can see how \(\theta_1\) is equivalent to both \(-24°\) or \(336°\). Since \(24°\) represents the acute angle between the \(x\)-axis and the arm between both arms, we can add this value to \(180°\) in order to get \(\theta₂\) (\(204°\)).