Basic Trigonometry

Introduction to Angles

Trigonometry is the study of angles and relationships between them. Especially important in trigonometry are the angles of a triangle. For this reason, trigonometry is closely linked with geometry. One of the major differences between trigonometry and geometry, though, is that trigonometry concerns itself with actual measurements of angles and sides of a triangle, whereas geometry focuses on establishing relationships between unmeasured angles and sides. To begin our study of trigonometry, we'll review the definition and some characteristics of angles to make sure we have a solid foundation for learning more about them.

Angles, by definition, lie in a plane, so trigonometry is a two-dimensional field of study. It will be convenient, and eventually necessary, to become familiar with the coordinate plane, which is a system of measuring and plotting points in two dimensions. The location of any point in a plane, then, can be specified by exact coordinates. A point can also be specified by a vector. A vector is like a line segment lying in a specific position--it has length and direction. Vectors can be used to determine the location of points, as well as the measure of certain angles. These basic concepts will provide a foundation for understanding the principles of trigonometry.

  

Angles Defined 

An angle is the union of two rays that share a common endpoint. The rays are called the sides of the angle, and the common endpoint is the vertex of the angle. The measure of an angle is the measure of the space between the rays. It is the direction of the rays relative to one another that determine the measure of an angle.

In trigonometry, angles are often defined in terms of rotation. Consider one ray, and then let it rotate a fixed distance about its endpoint. The ray in its initial position before the rotation, and the ray in its ending, or terminal position, after the rotation, creates an angle. The endpoint point about which the ray rotates is the vertex. The amount of rotation determines the measure of the angle. The ray in the initial position, before the rotation, is called the initial side of the angle. The ray in the terminal position, after the rotation, is called the terminal side of the angle. An angle created this way has a positive measure if the rotation was counterclockwise, and a negative measure if the rotation was clockwise.

Figure %: An angle defined as the rotation of a single ray

Note that a ray can rotate all the way around to its initial position, and possibly further, and still result in an angle with an initial and terminal side. This definition of an angle places no restriction on the magnitude of an angle (how far it can rotate). 

Measuring Angles

There are three units of measure for angles: revolutions, degrees, and radians. In trigonometry, radians are used most often, but it is important to be able to convert between any of the three units.

Revolutions

A revolution is the measure of an angle formed when the initial side rotates all the way around its vertex until it reaches its initial position. Thus, the terminal side is in the same exact position as the initial side. In trigonometry, angles can have a measure of many revolutions--there is no limit to the magnitude of a given angle. A revolution can be abbreviated "rev".

Degrees

A more common way to measure angles is in degrees. There are 360 degrees in one revolution. Degrees can be subdivided, too. One degree is equal to 60 minutes, and one minute is equal to 60 seconds. Therefore, an angle whose measure is one second has a measure of degrees. When perpendicularity is discussed, it is most often defined as a situation in which a 90 degree angle exists. Often degrees are used to describe certain triangles, like 30-60-90 and 45-45-90 triangles. As previously mentioned, however, in most cases that concern trigonometry, radians are the most useful and manageable unit of measure. Degrees are symbolized with a small superscript circle after the number (measure). 360 degrees is symbolized 360 ^o .

Radian

A radian is not a unit of measure that is arbitrarily defined, like a degree. Its definition is geometrical. One radian (1 rad) is the measure of the central angle (an angle whose vertex is the center of a circle) that intercepts an arc whose length is equal to the radius of the circle. The measure of such an angle is always the same, regardless of the radius of the circle. It is a naturally occurring unit of measure, just like Π is the natural ratio of the circumference of a circle and the diameter. If an angle of one radian intercepts an arc of length r , then a central angle of 2Π radians would intercept an arc of length 2Πr , which is the circumference of the circle. Such a central angle has a measure of one revolution. Therefore, 1 rev = 360 ^o = 2Π rad . Also, 1 rad = () ^o = rev.

Conversion between Revolutions, Degrees, and Radians

Below is a chart with angle measures of common angles in revolutions, degrees, and radians. Any angle can be converted from one set of units to another using the definition of the units, but it will save time to memorize a few simple conversions. It is particularly important to be able to convert between degrees and radians.

Figure %: Some common angles measured in all three units of measurement

Figure %: The coordinate plane

The Coordinate Plane

Angles lie in a plane. To specify the point in space where an angle lies, or where any figure exists, a plane can be assigned coordinates. Since a plane is two-dimensional, only two coordinates are required to designate a specific location for every point in the plane. One coordinate determines the length, and the other determines width. In reality, length and width are the same thing--they are used because they describe distance in two directions which are perpendicular to each other. This is all the coordinate plane is: a plane with two perpendicular axes by which distance in either of two dimensions can be measured.
The coordinate plane consists of an origin and two axes. The origin is a point. The axes are lines perpendicular to each other that intersect at the origin. Below is pictured the coordinate plane, with the origin at point O.
The origin is fixed, and designated as the point (0,0). Every other point is assigned an ordered pair, (x, y) , according to its position relative to the origin. The two axes are named the $x$-axis and the y-axis. In most drawings, the x -axis is the horizontal axis, and the y-axis is the vertical axis, but this does not necessarily need to be the case. A point is assigned an ordered pair consisting of two real numbers: The first is the x-coordinate, which measures how far the point is from the y-axis. The second real number making up an ordered pair is the y-coordinate, which measures the distance between the point and the x-axis. Often the axes are pictured with tick marks indicating length to make it easier to measure distance. When a point is drawn into the coordinate plane and assigned an ordered pair, it is plotted. Take a gander at the plotted points below.

Figure %: Some points plotted in the coordinate plane

Note that some of the coordinates are negative numbers. Negative distance does not exist, but coordinates are given either positive or negative values to specify which side of the given axis they are on. In most cases, the positive direction of the x-axis points to the right, and the positive direction of the y-axis points upward. Thus, for example, points on the left of the y-axis have a negative x-coordinate. The positive directions don't always have to be these directions, though. Often, as in the diagram above, the axes will only have an arrow on the end which points in the positive direction. The other end has no arrow. This is how one can tell where the positive and negative values lie.
A plane extends in all direction without limit. So does the coordinate plane. Although there are many ways to draw the coordinate plane, it is always the same thing: a point of origin and two axes, which intersect at the origin and lie perpendicular to each other. The origin, by definition, always has the coordinates (0,0). Every other point in the plane can be measured according to the axes. Even the point (33563452143,23455434) exists and can be located in any coordinate plane; it extends without limit. Below are some other ways to draw the coordinate plane. All look different, but they are all the same coordinate plane.

Figure %: Different views of the coordinate plane

The axes of the coordinate plane divide the plane into four regions--these regions are called quadrants. The region in which the x-coordinate and the y-coordinate are both positive is called Quadrant I. Quadrant II is the region in which x < 0 and y > 0 . Quadrant III is the region in which x < 0 and y < 0 . Quadrant IV is the region in which x > 0 and y < 0 . The quadrants are labeled in the figure below.

Figure %: Quadrants I - IV. 

Terms

Angle  -  The union of two rays with a common vertex.

Coordinate Plane  -  Any plane with two perpendicular intersecting lines. The point of intersection of the lines is called the origin and the location of any point in the plane can be measured along the two lines, or axes.

Degree  -  A unit of measure for angles. 360 degrees equals one revolution and 2Π radians.

Initial Side  -  The side of an angle from which the rotation begins; the initial position of the ray whose rotation creates the angle.

Magnitude  -  The length of a vector

Minute  -  A subdivision of a degree. One minute is equal to degrees.

Ordered Pair  -  The $x$-coordinate and $y$-coordinate, placed together between parentheses and separated by a comma. An ordered pair specifies a location in the coordinate plane.

Origin  -  The intersection of the $x$-axis and $y$-axis in a coordinate plane. The location of the origin is (0,0).

Plot  -  To draw a point in the coordinate plane at a specific location. Points are plotted in the coordinate plane.

Quadrant  -  One of the four regions in the coordinate plane created by the intersection of the axes

Quadrantal Angle  -  An angle in standard position whose terminal side lies along one of the axes

Radian  -  A unit of measure for angles. 2Π radians equals one revolution and 360 degrees.

Ray  -  A part of a line with a fixed endpoint on one end that extends without bound in the other direction.

Revolution  -  A rotation of the magnitude such that the initial side of an angle coincides with the terminal side; one complete rotation. One revolution equals 360 degrees and 2Π radians.

Second  -  A subdivision of a degree. One second is equal to degrees, or minutes.

Side  -  One of the rays that makes up an angle (is a side of the angle).

Standard Position  -  The location of an angle such that its vertex lies at the origin and its initial side lies along the positive $x$-axis.

Terminal Side  -  The side of an angle after rotation; the final position of the ray whose rotation created an angle.

Vector  -  A line segment with a starting point and an endpoint that represents motion in the direction of the endpoint, and which lies in a specific position such that its direction is fixed.

Vertex  -  The common endpoint of two rays that form an angle

x-axis  -  One of the two perpendicular lines that form the coordinate plane; usually the x-axis lies in a horizontal position.

x-component  -  The magnitude of a vector in the x direction.

x-coordinate  -  The distance between a point and the y-axis.

y-axis  -  One of the two perpendicular lines that form the coordinate plane; usually the y-axis lies in a vertical position.

y-component  -  The magnitude of a vector in the y direction.

y-coordinate  -  The distance between a point and the x-axis.

REFERENCES: sparknotes, Trigonometry Book-William Hart