## Angles and their Measures

Trigonometry is a branch of mathematics that studies triangles, their properties and applications to mathematical and real world problems. Since one of the basic parts of a triangle are its angles, then it will be the subject of the first module of this course.
CAROLINE D.C. JOSE
Posts: 8
Joined: Fri May 03, 2019 9:16 am

### Angles and their Measures

Trigonometry is a branch of mathematics that studies triangles, their properties and applications to mathematical and real world problems. Since one of the basic parts of a triangle are its angles, then it will be the subject of the first module of this course.

At the end of this module you are expected to:
1. Define the initial and the terminal side of an angle in Trigonometry
2. Name angles using proper designation symbols
3. Know the units of measurements used for angles
4. Convert from one unit of angle measurements to another
Initial and Terminal Sides / Rotation of Terminal Sides

In Geometry, an angle is defined as a union of two noncollinear rays that share a vertex. This definition is useful enough for the purposes of Geometry. In trigonometry, an angle is a figure formed by two rays that have a common endpoint such that one of the rays is a tactic ray (called the initial side) and the other ray is dynamic (or revolving) ray (called the terminal side). This is shown in the figure below.

The terminal side can either rotate clockwise or counterclockwise. The measure of an angle depends on the direction of the rotation of the terminal side: A counterclockwise rotation gives a positive measure while a clockwise rotation gives a negative measure. Thus, the angle shown above has a positive measure while the angle shown below has a negative measure.

An angle a vertex the origin and an initial side on the positive x-axis is said to be in the standard position. An angle that is in standard position is said to be located in the quadrant where its terminal side is. The Angle shown below is an angle in the standard position and is located in the fourth quadrant.

An angle in the standard position that has a terminal side on any of the two axes (x-axis or y-axis) is called a quadrantal angle. An example is shown below.

Angles in the standard position that have coincident terminal sides are called coterminal angles.

Symbols to Designated Angles
Since we will using angles a lot of times throughout the course, then it will be convenient if we have a way of naming or referring to angles.
In naming angles, we use the angle symbols (∠) followed by the name of the angle. There are many ways of naming angles. We will discuss the commonly used ways here.
Using Three Points

One way of naming angles is by using three points on the angle, one of which is the vertex. This is demonstrated in the examples below.

Note that in this way of naming angles, the middle letter must be the name of the vertex of the angle. The angle on the left above may not be named ∠ABC.
Using the Vertex
Another way of naming an angle by using the name of the vertex. This is particularly useful when naming an angle that is not adjacent to other angles.

Using Numbers
When naming several adjacent angles, using numbers can be convenient.

Using Letters
Instead of numbers, Greek letters or lowercase letters may also be used

Measurement of Angles
Aside from being able to name angles, it is also important that we can measure angles. Let us discuss the different units of measurement for angles.

Revolutions
One way of measuring angles is by the number of times the terminal side outlines a circle on the Cartesian plane. Look at the example below. The circle used has its center at the origin and has a radius or r.
The measure in revolutions of the angle shown at the left can be computed by getting the ratio of the length (s) of the intercepted arc to the circumference of the circle.
Revolutions (revs) = s/2πr

Degrees
The measure degree is obtained by dividing a circle into 360 parts. One such part is called a degrees of an angle can be related to the number of revolutions using the formula
Degrees = (number of revolutions)(360)
Using the conversation above, half a revolution would be equivalent to 180 degrees.

Radians
This measure is what we will be using frequently in this course. To understand the radian measure, note that 360 degrees is equivalent to 2π radians.

Radians = (number of revolutions)(2π/360)
Or, after simplifying
Radians =(number of degrees)( π/180)

Example: What is the measure of a 30 degree angle in radians?

Solution:
Since we know the degree measure of the angle. We can use conversion

Radians =(number of degrees)( π/180)
Radians = (30)( π/180) = π/6

Therefore, 30 degrees= π/6 radians

Example: An angle in the standard position is formed through 5 ½ revolutions of the terminal side. What is the measure of the angle in radians?

Solutions:
We can use the formula
Radians = (number of revolutions)(2π)
Radians = (5 ½)(2π) = 11π

Therefore, 5 ½ revolutions is equivalent to 11π radians.

CAROLINE D.C. JOSE
Posts: 8
Joined: Fri May 03, 2019 9:16 am

### Re: Angles and their Measures (Assignment and Quiz)

Which angle is coterminal with 150°?
b. 510°

8π radians = ?
d. 4 revs

How many revolutions are in 900°?
d. 2.5

Which of the following radian measures is equal to 300°?
a. 5π/3

How many radians are in 3.5 revolutions?
d. 7π

If the initial side of an angle is at QII and its terminal side got to the third quadrant
by rotating in the counterclockwise direction, then which of the following statements is TRUE about the angle?
a. Its smallest possible measure is less than 180°

Which of the following is the smallest angle measure?
b. 1 degree

What kind of angle is an angle that has an initial side at the positive x-axis and a terminal side at the negative y-axis?
c. Quadrantal

Which of the following angle measure is the largest?
a. 5π/6 Rads

Which of the following degree measures is equivalent to 3π radians?
d. 540°

A and B are coterminal angles. If the measure of B is 910°, then in what quadrant is A located?
c. QIII

What must we do to the degree measure of an angle to convert it to radian measure?
b. Multiply it by π/180

An angle in the standard form has an initial side at the positive x-axis.
True

An angle in the standard position with a measure of –790° is located in which quadrant?
c. QIV

A quadrantal angle is an angle in the standard form and has a terminal side in either axis.
True

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