Right Triangle Trigonometry
7.1 Angles, Radians, Arc Length, and Angular Speed
What Is an Angle?
Trigonometry is really the study of how angle measure controls shape, distance, and motion. So before we do any heavy computation, we need a very grounded picture of what an angle actually is and how we place it in the coordinate plane.
Definition: Ray, Vertex, and Angle
A ray is a part of a line that starts at one point and continues forever in one direction.
An angle is formed by two rays that share a common endpoint.
- The common endpoint is the vertex.
- The rays are the sides of the angle.
Definition: Standard Position
An angle is in standard position when:
- its vertex is at the origin, and
- its initial side lies on the positive x-axis.
The other side is called the terminal side.
Positive angles rotate counterclockwise. Negative angles rotate clockwise.
Fact: Measuring an Angle
Conventionally, there are $360^\circ$ in a circle.
Example: Determine sign from direction
Suppose an angle starts on the positive x-axis and rotates 70 degrees clockwise. Express it with sign.
Show Solution
Clockwise rotation means a negative angle.
So the angle measure is:
$$ -70^\circ $$
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Pi and Radians
The choice to use $360^\circ$ as a way to measure angles is largely arbitrary. We could have chosen anything, for example: why not $100^\circ$ for the full circle? The answer is that $100$ isn't quite as nice for cutting up.
Consider the divisors of 100. We can cut 100 up in 1 piece of 100, or 2 pieces of 50, or 25 pieces of 4 etc. There are a few ways. But, we wanted to cut 100 into 3 pieces, it can't be done. $\frac{100}{3}=33.333...$. $360$ on the other hand has many divisors.
| Divisors of 100 | Pair | Divisors of 360 | Pair |
|---|---|---|---|
| 1 | 100 | 1 | 360 |
| 2 | 50 | 2 | 180 |
| 4 | 25 | 3 | 120 |
| 5 | 20 | 4 | 90 |
| 10 | 10 | 5 | 72 |
| 6 | 60 | ||
| 8 | 45 | ||
| 9 | 40 | ||
| 10 | 36 | ||
| 12 | 30 | ||
| 15 | 24 | ||
| 18 | 20 |
We can cut 360 up into 3's, 6's, 9's, and many other ways that 100 can't be cut into.
Because this choice is largely arbitrary, mathematicians have chosen to not use just some number to measure angles, but a number intrinsic to the circle. It turns out that no matter the size of the circle, if you take its diameter and ask, how many of these diameters does it take to wrap around the circle, the answer is always the same. It's around $3.14159\ldots$.
Definition: Pi and Radians
A radian is a unit of measurement that measures the angle central to a circle that is subtended by an arc that is the radius of the circle.
It takes $2\pi$ radians to wrap all the way around the circle where $\pi$ is the ratio of circumference (C) to the diameter (D) of a circle: $\pi=\frac{C}{D}$. $\pi$ is irrational and is around $3.14159$.
Because a radian is defined as $\theta=\frac{\text{arc length}}{\text{radius}}=1$, the units technically cancel. So numerically speaking, a radian does not have a unit like degrees. We will sometimes still write "rad" or "radians" to distinguish the angle from other plain numbers.
Fact: Radians in a Circle
It takes $2\pi$ radians to completely go around the circle.
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Quadrants, Coterminal Angles, and Quadrant Angles
Now that we can place an angle, the next move is classifying where it lands. This is where quadrants and coterminal angles become a huge simplifier.
Definition: Quadrants
The coordinate plane is split into four quadrants, numbered counterclockwise:
- Quadrant I (upper right)
- Quadrant II (upper left)
- Quadrant III (lower left)
- Quadrant IV (lower right)
Fact: Coterminal Angles
Angles are coterminal when they share the same terminal side.
In degrees: $$ \theta_{\text{coterminal}} = \theta + 360^\circ k, \quad k \in \mathbb{Z} $$
In radians: $$ \theta_{\text{coterminal}} = \theta + 2\pi k, \quad k \in \mathbb{Z} $$
Definition: Quadrant Angles
A quadrant angle has terminal side on an axis.
Common examples: $$ 0^\circ,90^\circ,180^\circ,270^\circ $$ or $$ 0,\frac{\pi}{2},\pi,\frac{3\pi}{2} $$
Example: Determine quadrant
Determine where each angle lands:
- $150^\circ$
- $-78^\circ$
- $480^\circ$
- $\frac{19\pi}{18}$
Show Solution
- $150^\circ$ is between $90^\circ$ and $180^\circ$, so Quadrant II.
- Add $360^\circ$ to locate it in $[0,360)$:
$$ -78^\circ + 360^\circ = 282^\circ $$
Since $282^\circ$ is between $270^\circ$ and $360^\circ$, it is in Quadrant IV.
- Subtract $360^\circ$:
$$ 480^\circ - 360^\circ = 120^\circ $$
$120^\circ$ is in Quadrant II.
- Compare with key radian boundaries:
$$ \pi = \frac{18\pi}{18}, \quad \frac{3\pi}{2} = \frac{27\pi}{18} $$
Since $$ \frac{19\pi}{18} $$ is between $\pi$ and $\frac{3\pi}{2}$, it is in Quadrant III.
Example: Find coterminal angle in a target interval
Find an angle coterminal with $\frac{5\pi}{3}$ in the interval $[4\pi,6\pi)$.
Show Solution
Start from: $$ \theta = \frac{5\pi}{3} + 2\pi k $$
Write everything over denominator 3: $$ \theta = \frac{5\pi}{3} + \frac{6\pi k}{3} = \frac{(5+6k)\pi}{3} $$
We need: $$ 4\pi \le \theta < 6\pi $$
Multiply by 3 and divide by $\pi$: $$ 12 \le 5+6k < 18 $$
Subtract 5: $$ 7 \le 6k < 13 $$
So $k=2$ is the only integer that works. Then: $$ \theta = \frac{5\pi}{3} + 2\pi(2) = \frac{5\pi}{3}+\frac{12\pi}{3} = \frac{17\pi}{3} $$
Answer: $$ \frac{17\pi}{3} $$
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Radians and Angle Conversion
Degrees are familiar, but radians are the natural language of circles. Most trig formulas are cleaner in radians, so this is where things start to feel really trigonometric. In many cases we will need to convert from radians to degrees, or vice versa.
Fact: Conversion Formulas
Use these two conversions:
$$ \text{degrees} \to \text{radians}:\quad \theta_{\text{rad}} = \theta_{\deg}\cdot \frac{\pi}{180} $$
$$ \text{radians} \to \text{degrees}:\quad \theta_{\deg} = \theta_{\text{rad}}\cdot \frac{180}{\pi} $$
| Degrees | Radians |
|---|---|
| $0^\circ$ | $0$ |
| $30^\circ$ | $\frac{\pi}{6}$ |
| $45^\circ$ | $\frac{\pi}{4}$ |
| $60^\circ$ | $\frac{\pi}{3}$ |
| $90^\circ$ | $\frac{\pi}{2}$ |
| $120^\circ$ | $\frac{2\pi}{3}$ |
| $135^\circ$ | $\frac{3\pi}{4}$ |
| $150^\circ$ | $\frac{5\pi}{6}$ |
| $180^\circ$ | $\pi$ |
| $210^\circ$ | $\frac{7\pi}{6}$ |
| $225^\circ$ | $\frac{5\pi}{4}$ |
| $240^\circ$ | $\frac{4\pi}{3}$ |
| $270^\circ$ | $\frac{3\pi}{2}$ |
| $300^\circ$ | $\frac{5\pi}{3}$ |
| $315^\circ$ | $\frac{7\pi}{4}$ |
| $330^\circ$ | $\frac{11\pi}{6}$ |
| $360^\circ$ | $2\pi$ |
Fact: Default Unit Convention
If no unit is shown in a trig expression, the default interpretation is radians.
Example: Convert degrees to radians
Convert each angle to radians:
- $155^\circ$
- $-270^\circ$
- $47^\circ$
Show Solution
- $155^\circ\cdot\frac{\pi}{180} = \frac{155\pi}{180} = \frac{31\pi}{36}$
- $-270^\circ\cdot\frac{\pi}{180} = -\frac{270\pi}{180} = -\frac{3\pi}{2}$
- $47^\circ\cdot\frac{\pi}{180} = \frac{47\pi}{180}$ (Already simplified since 47 and 180 share no common factor.)
Example: Convert radians to degrees
Convert each angle to degrees:
- $\frac{4\pi}{3}$
- $\frac{-11\pi}{6}$
- $2$ radians
Show Solution
- $\frac{4\pi}{3}\cdot\frac{180}{\pi} = 4\cdot60 = 240^\circ$
- $\frac{-11\pi}{6}\cdot\frac{180}{\pi} = -11\cdot30 = -330^\circ$
- $2\cdot\frac{180}{\pi} = \frac{360}{\pi} \approx 114.592^\circ$
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Degrees-Minutes-Seconds (DMS)
Just like the hours on a clock can be broken into minutes and seconds, so can the degrees in a circle. This is less common in algebra-heavy work, but it appears in navigation and astronomy, so it is still worth being fluent.
Definition: Degrees, Minutes, and Seconds
$$ 1^\circ = 60' = 3600'' $$ A single ' denotes minutes and the double '' denotes seconds.
Example: Convert DMS to decimal degrees
Convert $43^\circ 15' 17''$ to decimal degrees.
Show Solution
Write minutes and seconds as parts of a degree:
$$ 43^\circ 15' 17'' = 43 + \frac{15}{60} + \frac{17}{3600} $$
Compute each part: $$ \frac{15}{60}=0.25, \quad \frac{17}{3600}\approx 0.0047222 $$
Add: $$ 43 + 0.25 + 0.0047222 = 43.2547222^\circ $$
Example: Convert decimal degrees to DMS
Convert $18.355^\circ$ to DMS.
Show Solution
Separate whole degrees from decimal part: $$ 18.355^\circ = 18^\circ + 0.355^\circ $$
Convert decimal degrees to minutes: $$ 0.355\cdot 60 = 21.3' $$
So we have $21'$ and leftover $0.3'$.
Convert leftover minutes to seconds: $$ 0.3\cdot 60 = 18'' $$
Final answer: $$ 18^\circ 21' 18'' $$
If you're using a TI83/84 like I do for my classes, then you can find the angle menu just above the apps button (2nd + apps). From here you can find markers for these units and a DMS function which will convert a decimal to degrees, minutes, and seconds. For some undiscernible reason, the seconds symbol is not in the menu itself but is an alpha key located above the plus symbol on the TI-84 Plus CE.
| Angle Menu (2nd + Apps) | DMS Conversion Screen |
|---|---|
 |
 |
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Arc Length and Sector Area
Now we connect angle measure to actual distance and area on circles. This is where radians really pay off. If you want to take a piece of something, you can multiply by the size of the piece you want. For example, if you want half of $1000$, the of literally means to multiply. So $\frac{1}{2}\cdot 1000 = 500$ means half of $1000$ is $500$. Because of the incredible symmetry of the circle, we can also find pieces of its circumference. You may recall from earlier classes that the circumference of a circle is $C=2\pi r$. If we were to measure a piece of that circumference, we would be measuring an arc. Because this distance is measured in radians, we need only to specify how many radians of that circle we want and multiply it by the circumference.
Similarly, the piece of the area of a circle is called a sector.
Theorem: Arc Length and Sector Area Formula
If $\theta$ is measured in radians, then arc length is
$$ s = r\theta $$
and the sector area is
$$ A_{\text{sector}} = \frac{1}{2}r^2\theta $$
Example: Arc length from degree measure
Find the arc length on a circle of radius $10\text{ cm}$ subtended by $215^\circ$.
Show Solution
First convert angle to radians: $$ 215^\circ\cdot\frac{\pi}{180} = \frac{43\pi}{36} $$
Now apply $s=r\theta$: $$ s=10\cdot\frac{43\pi}{36} = \frac{430\pi}{36}=\frac{215\pi}{18}\text{ cm} $$
Approximate: $$ \frac{215\pi}{18}\approx 37.52\text{ cm} $$
Example: Sector area from radian measure
Find the area of a sector with radius $5\text{ cm}$ and angle $\frac{\pi}{3}$.
Show Solution
Use: $$ A_{\text{sector}}=\frac{1}{2}r^2\theta $$
Substitute: $$ A_{\text{sector}}=\frac{1}{2}(5)^2\left(\frac{\pi}{3}\right) $$
Simplify: $$ A_{\text{sector}}=\frac{25\pi}{6}\text{ cm}^2 $$
Approximate: $$ \frac{25\pi}{6}\approx 13.09\text{ cm}^2 $$
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Linear and Angular Speed
This is the motion version of everything above. We use it in wheels, gears, rotating platforms, and orbital models.
Definition: Angular and Linear Speed
If an object rotates through angle $\theta$ in time $t$:
$$ \omega = \frac{\theta}{t} $$ where $\omega$ is angular speed (rad/s).
If it travels arc length $s$ in time $t$:
$$ v = \frac{s}{t} $$ where $v$ is linear speed.
Since $s=r\theta$, we get $$ v=r\omega $$
Example: Angular speed from rotations
A wheel makes 1 full rotation every 5 seconds. Find angular speed in rad/s.
Show Solution
One full rotation is: $$ 2\pi\text{ radians} $$
Time is $5$ seconds, so: $$ \omega = \frac{\theta}{t}=\frac{2\pi}{5}\text{ rad/s} $$
Approximate: $$ \omega\approx 1.257\text{ rad/s} $$
Example: Convert rpm to linear speed
A bike wheel has diameter $22$ inches and rotates at $140$ rpm. Find linear speed in mph.
Show Solution
Radius: $$ r=\frac{22}{2}=11\text{ in} $$
Convert rpm to rad/min: $$ \omega = 140\cdot 2\pi = 280\pi\text{ rad/min} $$
Use $v=r\omega$ (inches per minute): $$ v = 11\cdot 280\pi = 3080\pi\text{ in/min} $$
Convert to miles/hour: $$ 3080\pi\frac{\text{in}}{\text{min}}\cdot\frac{60\text{ min}}{1\text{ hr}}\cdot\frac{1\text{ ft}}{12\text{ in}}\cdot\frac{1\text{ mi}}{5280\text{ ft}} $$
Simplify constants: $$ = \frac{3080\pi\cdot 60}{12\cdot 5280}\text{ mph} = \frac{385\pi}{132}\text{ mph} $$
Approximate: $$ \frac{385\pi}{132}\approx 9.16\text{ mph} $$