Angles in Trigonometry | The Unit Circle |

Degrees, Radians, and Co-Terminal Angles | More Practice |

## Angles in Trigonometry

Even though the word trigonometry is derived from the word “triangle”, you’ll see a lot of **circles** when you work with Trig! We talked about **angle measures** in the **Right Triangle Trigonometry **section, and now we’ll see how angles relate to the **circumference of a circle**.

Again, an angle is made up of **two rays**. A ray is a line that extends forever starting at a point called a **vertex**. Think of the **initial side ray** as the ray where the angle **starts**, and the **terminal side ray** as the ray where the angle **stops**.

An angle is in **standard position** if its vertex is at the origin $ \left( {0,0} \right)$ and its initial side starts at the** positive **$ x$**-axis**. **Quadrants** are the four different areas of the coordinate system, and “**Quadrant I**” (“Quadrant One” ) starts where both $ x$ **and **$ y$ **are positive**. It helps me to remember where the quadrants are if you draw a big “**C**” across the coordinate system:

**Quadrant I** is between **0° and 90°**, **Quadrant II** is between **90° and 180°**, **Quadrant III** is between **180° and 270°**, and **Quadrant IV** is between **270° and 360°** (back to **0°**).** **Don’t worry why – someone just decided it would be this way.

Angle measurements **start at 0** on the positive $ x$-axis and go **counter-clockwise** around the “circle” until they are **back at 360**, as shown in the graphs below. Remember from** Geometry **that there are** 360 degrees (**

**. When going**

**360°)****in a circle****clockwise**from the positive $ x$-axis, angles measurements are

**negative**.

Here are some examples. **Note that these graphics are extremely important to understand how angles are set up in Trigonometry:**

Positive Angles | Negative Angles |

## Degrees, Radians, and Co-terminal Angles

### Degrees

Again, since there are **360°**** in a circle**; it makes sense that an angle that is formed by rotating the terminal side **360°** actually coincides with itself; in other words, it’s the same angle. We call these angles like this **co-terminal**, since they have the same angle measurement, starting from **0°**.

**One degree** is $ \displaystyle \frac{1}{{360}}$ of a complete revolution, and thus, as we saw above, a right angle pointing up is **90°**, a straight angle pointing to the left is **180°**, and a right angle pointing down is **270°**.

### Converting Decimal Degrees to/from Degrees, Minutes and Seconds

Sometimes you’ll have to know how to convert **degrees in decimals** to **degrees**,** minutes**, and** seconds** (a measure dating back to the Babylonians, and is used in latitudes and longitudes). To do these conversions, multiply or divide by powers of **60**, since there are **60** **seconds in a minute** and **60** **minutes in a degree** (thus **3600** seconds in **1** degree). Here are some examples:

Convert **48.78° ****to degrees, minutes, and seconds: **The “whole” part of the degrees is **48**, so we have **48°. ** Next, **.78 **is leftover, which is $ \displaystyle .78\times 60=46.{8}’$ (minutes). Then, **.8** is leftover, which is $ .8\times 60=48″$ (seconds). Thus, we have $ 48.78{}^\circ =48{}^\circ \,46’\,48″$.

To do this conversion on the **graphing calculator** (make sure in mode is **DEGREE**), type in **48.78**, then hit **2**^{nd} apps (angle), then hit **4** or scroll to DMS, and hit **ENTER**: .

Convert $ 48{}^\circ \,46’\,\,48″$** **to **decimal degrees**: When converting back to degrees in decimal, divide the minutes by **60** and the seconds by **3600** and add to the whole degrees: $ \displaystyle 48{}^\circ \,46’\,48″=48+\frac{{46}}{{60}}+\frac{{48}}{{3600}}=48.78{}^\circ $**.**

To do this conversion on the **graphing calculator** (make sure in mode is **DEGREE**), type in **48** for degrees, then hit **2**^{nd} apps (angle)** 1**. Then type in **46** for minutes and hit **2**^{nd} apps (angle) **2**. Then type in **48** for seconds and hit alpha + (seconds), and hit **ENTER**: .

### Radians

As you get more advanced in trig, you’ll deal with **radians **instead of degrees, and these can be confusing.

First some definitions: if you were to create a circle around the origin, a **central angle** is the angle formed by having the vertex at the origin, and the **arc** **length** of that circle is the measurement around the **circumference** of that circle (like if you were to take measuring tape around the circle).

The word **radian** comes from the word **radius**, and is just another way to measure angles in a “trig circle”. **One radian** means that the **length of the radius of that circle is the same as the arc length**; and we like to deal with “trig circles” where the radius is **1**.

The circumference of a circle is $ 2\pi r$ so when the radius is **1** in a “trig circle”, all the way around is $ 2\pi \left( 1 \right)$** **or $ 2\pi$ radians. This circle is called a **Unit Circle**; you will get intimately involved with this circle ;). $ 2\pi$ is about **6.28** (**2** times **3.14**).

Again, the coordinate system consists of four **quadrants**. **Quadrant I** is between **0** and $ \displaystyle \frac{\pi }{2}$ radians, **Quadrant II** is between $ \displaystyle \frac{\pi }{2}$ and $ \pi$ radians, **Quadrant III** is between $ \pi $* *and $ \displaystyle \frac{3\pi }{2}$ radians, and **Quadrant IV** is between $ \displaystyle \frac{3\pi }{2}$ and $ 2\pi$ radians (back to **0** radians).** **

The main reason we want to work with **radians** and not **degrees** is that degrees are sort of arbitrary, whereas radians have real meaning since they are tied to the circumference of a circle. There are many other reasons, too, since it’s much easier to do calculations with radians than degrees in **Calculus**.

The weird thing about radians is that they really **don’t have a unit**, like degrees, feet or meters. They are just radians.

Here are some graphics of the concept of radians, which may help:

Concept of Radians | |

### Co-Terminal Angles

We saw earlier that a complete revolution of the “trig circle” is **360°** or $ 2\pi $ **radians**. If we are given an angle that is greater than either **360°** or $ 2\pi $ radians (either in positive or negative measurements), we have to keep subtracting (or adding, if we have a negative angle) either **360** or $ 2\pi$ until we get an angle between **0** and **360°**, or **0** and $ 2\pi$ radians. We call the new angle **co-terminal** with the original angle, since they are the same angle in the “trig circle”.

Let’s do some problems. Notice that when dealing with radians, we have to remember how to get and use **Greatest Common Denominators to Add Fractions**. Don’t worry; all of this will get easier!

Here are more types of problems you may see with co-terminal angles:

### Converting Degrees to Radians, and Radians to Degrees

You’ll have to know how to **convert back and forth between degrees and radians**. Since we saw above that there are $ 2\pi$ radians in **360°**, we know that there are $ \pi $ radians in **180°**. Thus, when doing the conversions between degrees and radians, we are always multiplying by either $ \displaystyle \frac{\pi }{180}\,\,\text{or }\frac{180}{\pi }\text{ }$. You can see this by converting **90 degrees** to radians by using a **Unit Multiplier: **

$ \displaystyle \begin{array}{l}\displaystyle \frac{{90\text{ }\cancel{{\text{degrees}}}}}{1}\,\,\times \,\,\displaystyle \frac{{\pi \text{ radians}}}{{\text{180 }\cancel{{\text{degrees}}}}}\,\,\\\,\,\,\,\,\,\,\,\,\,\,=\,\,\displaystyle \frac{{90\pi \text{ radians}}}{{180}}\,\,=\,\,\displaystyle \frac{\pi }{2}\,\,\text{radians}\end{array}$

**Here is a trick**: If you are converting from degrees to radians, you want a $ \pi$ in the answer, so the $ \pi$ should be on the top of the fraction (so multiply by $ \displaystyle \frac{\pi }{{180}}$). If you are converting from radians to degrees, you want to get rid of the $ \pi$, so the $ \pi$* *should be on the bottom of the fraction (so multiply by $ \displaystyle \frac{{180}}{\pi }$).

Note that you can convert back and forth from degrees to radians in the **graphing calculator**. To convert from degrees to radians, put the calculator in **RADIAN** mode (mode – RADIAN), then type in the angle in degrees. Then hit **2**^{nd} apps (angle) **1** (degrees), and hit **ENTER**:

To convert back to degrees, put the calculator in **DEGREE** mode (mode – DEGREE), then type in the angle in radians (using parentheses, if necessary). Then hit **2**^{nd} apps (angle) **3** (radians), and hit **ENTER: **

Let’s do some problems (without using graphing calculator conversions!):

## The Unit Circle

The **Unit Circle** is probably the most important tool you’ll use in both **Pre-Calculus**, and then later (occasionally) in **Calculus**. The Unit Circle is basically a visual representation of certain “**special angles**”, for which the **exact values of the trig functions** are known. It is called the “unit” circle, since its **radius is 1**.

The reason you’ll have to “memorize” the Unit Circle is so that you can come up with trig values for these angles quickly and without a calculator. That’s all it is!

Here it is! For extra practice, there are good printouts of the Unit Circle here on the **Embedded Math** website, including blank ones you can practice filling in.

**Note**: Sometimes you will see $ \displaystyle \frac{1}{\sqrt{2}}$ instead of $ \displaystyle \frac{\sqrt{2}} {2}$ in the Unit Circle. This is the same number; the former has not been rationalized, and the latter has ben rationalized (see how to rationalize fractions with radicals in the **Powers, Exponents, Radicals and Scientific Notation** Section here).

Here’s how it works (and we’ll show why below with an example): For the ordered pairs on the outside, the first number ($ x$ value) is the **cosine** of the angle, and the second ($ y$ value) is the **sine** of the angle. If you were to divide the second by the first (second over first), you’ll get the **tangent** of the angle.

Before I give you hints on how to learn the Unit Circle, let’s see how we arrived at the order pairs by demonstrating **Right Triangle Trigonometry. **We know from either a **30-60-90 triangle** (from **Geometry**), using our calculator, or using the **Pythagorean Theorem**, that the sides for the triangle below are **1** for the hypotenuse (since it’s a Unit Circle), $ \displaystyle \frac{1}{2}$ for the shortest side or leg, and $ \displaystyle \frac{{\sqrt{3}}}{2}$ for the longer leg. Thus we can “prove” the sin and cos values for **30°**. All the other special angles have similar proofs.

## Reciprocal Trig Functions

To get the other three trigonometric values** (cosecant **or** csc, secant **or** sec**, and **cotangent **or** cot**), just take the **reciprocals** (flip) of the **sin**, **cos,** and **tan** values, respectively.

For example, the **csc** of **30°** is $ \displaystyle \frac{1}{{\frac{1}{2}}}=2$, the **sec **of **30°** is $ \displaystyle \frac{1}{\frac{\sqrt{3}}{2}}\,=\,\frac{2}{\sqrt{3}}\,=\,\frac{2\sqrt{3}}{3}$, and the **cot **of **30°** is $ \displaystyle \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}$.

## Hints of Learning Unit Circle

I highly recommend that you print out a lot of blank Unit Circle charts (again, here on the **Embedded Math **website) and fill them out until you can do it without thinking :). Also, to get the other three trigonometric values (**csc**, **sec**, and **cot** values), just take the reciprocals of the **sin**, **cos** and **tan** values, respectively.

Here are additional hints:

- Notice that if you know the
**(cos, sin)**ordered pair values in the first quadrant, you know them in all the quadrants! Look for the mirror images of the ordered pairs, but with the**different signs**for that quadrant. I like to use the mnemonic**A**ll**S**tudents**T**ake**C**alculus, as shown in the Unit Circle chart, to remember the signs: starting in the first quadrant,**ALL**trigonometric values are**positive**, in the second quadrant,**Sin**(and thus**csc**) values are**positive**, in the third quadrant,**Tan**(and thus**cot**) values are**positive**, and in the fourth quadrant,**Cos**(and thus**sec**) values are**positive**. - Notice that the denominators of the three inside radian measurements are “
**3 4 6**”, in every quadrant (with the “**6**” closest to the horizontal axis, and a $ \pi $ in the numerator). For the first quadrant, there is**nothing**in the numerator except for the $ \pi $, in the second quadrant the numerator is**one less**than the denominator, in the third quadrant, the numerator is**one more**than the denominator, and in the fourth quadrant, the numerator is**twice**the denominator minus**1**. - Think of different slices of the circle, like slices of a pie. When dividing up the circle into
**fourths**, think of going around starting with**0**, and adding**90°**(or $ \displaystyle \frac{\pi }{2}$) counter-clockwise. If we have $ \left( {0,0} \right)$ in the center, we have $ \left( {1,0} \right)$, $ \left( {0,1} \right)$, $ \left( {-1,0} \right)$, and $ \left( {0,-1} \right)$ on the axes: . - When dividing the circle into
**sixths**, think about going around the circle counterclockwise, and adding**30°**, or $ \displaystyle \frac{\pi }{6}$ for radians ($ \displaystyle \frac{\pi }{6},\frac{{2\pi }}{6},\frac{{3\pi }}{6},\frac{{4\pi }}{6},\frac{{5\pi }}{6},$ … ). - When dividing the circle into
**thirds**, add**60°**, or $ \displaystyle \frac{\pi }{3}$ for radians: $ \displaystyle \frac{\pi }{3},\frac{{2\pi }}{3},\frac{{3\pi }}{3},\frac{{4\pi }}{3},\frac{{5\pi }}{3},\text{and}\,\frac{{6\pi }}{3}$. . - Try to visualize the triangles in the Unit Circle, and think where the placement of the
**(sin, cos)**coordinate points are. For example, for**30°**in the first quadrant, the $ x$-value is greater than the $ y$-value, so the coordinates of this point are $ \displaystyle \left( {\frac{{\sqrt{3}}}{2},\frac{1}{2}} \right)$, since $ \displaystyle \frac{{\sqrt{3}}}{2}>\frac{1}{2}$. (Of course, $ \displaystyle \frac{\sqrt{2}}{2}=\frac{\sqrt{2}}{2}$, so the coordinates for**45°**are the same.) Use this method to go around the triangle, and remember the coordinates of the other special angles by just keeping in mind the signs of the $ x$-values and $ y$-values: . - Some choose to remember the Unit Circle coordinates
**(sin, cos)**pairs by remembering**13-22-31**. Starting from the top right, if you remember**13-22-31**coming down to the $ x$-axis, you can remember $ \displaystyle \left( {\frac{1}{2},\frac{{\sqrt{3}}}{2}} \right),\left( {\frac{{\sqrt{2}}}{2},\frac{{\sqrt{2}}}{2}} \right),\left( {\frac{{\sqrt{3}}}{2},\frac{1}{2}} \right)$. See how you have**13-22-31**on the top of the fractions (with square roots), and**2**on the bottom? Then use the fact that the other coordinates are mirror images, making allowances for the different signs in each quadrant: . - If you have a
**negative angle**, just go forward to see how far you’ve gone, and then go that much the other way (clockwise) from**0**. For example, to get the**cos**of $ \displaystyle -\frac{{4\pi }}{3}$, you look to see how far forward you get to $ \displaystyle \frac{{4\pi }}{3}$, and then you go backwards (clockwise) that much to end up at $ \displaystyle \frac{2\pi }{3}$. Of course, you can always just add $ 2\pi $ to $ \displaystyle -\frac{4\pi }{3}$ to get $ \displaystyle \frac{2\pi }{3}$: .

## Finding Exact Values of Angles Using Unit Circle

Here are some problems that you may have to work, using what you know about co-terminal angles and the Unit Circle. Note that at first, you’ll want to redraw the whole Unit Circle if you have to know these for a test, and later, you can just draw part of the circle, since you’ll know it so well!

Now let’s work on getting **all six (6) trigonometric values** for the following special angles:

**Understand these problems, and practice, practice, practice!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the **Mathway** site, where you can register for the **full version** (steps included) of the software. You can even get math worksheets.

You can also go to the **Mathway** site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!

On to **Linear and Angular Speeds, Area of Sectors, and Length of Arcs** – you’re ready!