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About Trigonometry
TABLE OF CONTENTS
About Trigonometry ........................................................................................................... 1
What is TRIGONOMETRY? ......................................................................................... 1
Triangles ............................................................................................................................. 1
Background ..................................................................................................................... 1
Trigonometry with Triangles .......................................................................................... 1
Circles ................................................................................................................................. 2
Trigonometry with Circles.............................................................................................. 2
Rules/Conversion................................................................................................................ 3
Trigonometry Rules ........................................................................................................ 3
Converting Between Radians and Degrees..................................................................... 4
Functions............................................................................................................................. 4
Trigonometric Functions................................................................................................. 4
Graphing ............................................................................................................................. 6
Graphing Trigonometric Functions................................................................................. 6
Glossary .............................................................................................................................. 8
References......................................................................................................................... 10
About Trigonometry
What is TRIGONOMETRY?
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Trigonometry explores relationships between the sides and angles of a
triangle.
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Combining arithmetic, algebra, and geometry, trigonometry is used in
disciplines such as architecture and physics.
Triangles
Background
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Deriving its name from the Greek word for triangle, trigonometry
relates the measure of the sides of a triangle to the measure of its
angles. With this stated, it is important to note that triangles do not
hold a monopoly over trigonometric functions, as these functions can
also be defined using circles.
Trigonometry with Triangles
●
A right triangle is a triangle with a right angle (90 degrees or π/2
radians).
The Pythagorean Theorem is central to trigonometry. The theorem
states that for a right triangle with side lengths of a and b, and a
hypotenuse of length c, the following relationship always holds:
a2 + b2 = c2
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●
Trigonometric functions relate to angle measurement. The Greek
letter Ө is often used to represent the angle, as shown in the figure
below.
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The six trigonometric functions are defined as follows:
sin(Ө) = a/c = the side opposite the angle divided by the
hypotenuse
cos(Ө) = b/c = the side adjacent to the angle divided by the
hypotenuse
tan(Ө) = a/b = the side opposite the angle divided by the
side adjacent to the angle
sec(Ө) = c/b = the reciprocal of cos(Ө)
csc(Ө) = c/a = the reciprocal of sin(Ө)
cot(Ө) = b/a = the reciprocal of tan(Ө)
The standard abbreviations for the functions were used above, but the
full names are as follows: sine, cosine, tangent, secant, cosecant, and
cotangent.
Circles
Trigonometry with Circles
●
The trigonometric functions can be defined using a unit circle with
centre (0, 0). A unit circle is a circle with a radius of 1. For any point
(x, y) on the circle, the angle Ө is defined as the angle between the
positive x-axis and the line segment joining (0, 0) to that point.
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●
The trigonometric functions, as related to circles, are as follows:
sin(Ө) = y
cos(Ө) = x
tan(Ө) = y/x
sec(Ө) = 1/x
csc(Ө) = 1/y
cot(Ө) = x/y
To check that these are correct, simply draw a right triangle inside the
circle with sides x and y and hypotenuse 1.
Rules/Conversion
Trigonometry Rules
●
Recall that the Cartesian Coordinate plane contains 4 quadrants. The
CAST rule uses these quadrants to illustrate some helpful properties,
as follows:
Quadrant 4: Cos function
positive
Quadrant 1: All trigonometric
functions positive
Quadrant 2: Sin function
positive
Quadrant 3: Tan function
positive
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●
Another essential tool in remembering the properties of trigonometric
functions is that of the SOHCAHTOA rule, illustrated below.
Note that when you take out the first letter of each term in the
equations, you get the phrase "SOHCAHTOA".
Converting Between Radians and Degrees
●
The convention is that positive angles are measured counterclockwise
from the positive x-axis, and negative angles are measured clockwise
from the positive x-axis.
Note: π radians = 180 degrees
Functions
Trigonometric Functions
●
As it is geometrically impossible for a right triangle to contain any
obtuse angles, we turn to the circle definition, as it works for any
angle. To apply this, draw a right triangle within the circle, using the
ray that defines the angle and using the x-axis. In the first quadrant,
this is simply the right triangle containing the angle Ө. In the second
quadrant, it is the right triangle containing the supplementary angle
(180° - Ө or π - Ө), as illustrated below.
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●
In the third quadrant, the right triangle
contains the angle minus the straight
angle (Ө - 180° or Ө - π), as illustrated
in the figure to the right.
●
In the fourth quadrant, the right
triangle contains the conjugate angle
(360° - Ө or 2π - Ө), as illustrated in
the figure to the right.
●
The point connecting the circle to the ray is of the form
(x, y), where x = cos(Ө) and y = sin(Ө). Consider the figure below,
relating it back to the circle.
●
Provided below, for reference, is a chart containing the important
angles used in relation to trigonometric functions.
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Graphing
Graphing Trigonometric Functions
●
●
Graphs applicable to the trigonometric functions previously
considered, are provided below.
Sin
Cos
Tan
Csc
Sec
Cot
Consider the following:
- sin and cos are defined everywhere, while all other trigonometric
functions have asymptotes.
- The portion of the graph from -2π to 0 is the same shape as the
portion of the graph from 0 to 2π for sin, cos, sec, and csc
(e.g. cos(x) = cos(x + 2π)). This is due to the fact that the functions
are periodic, with a period of 2π. tan and cot are periodic, with a
period of π (e.g. tan(x) = tan(x + π)).
- sin and cos are the same shape, shifted 90° or π/2 radians
(i.e. sin(x) = cos(x - π/2)). Similarly, csc and sec are the same shape
shifted 90° or π radians (i.e. csc(x) = sec(x - π/2)).
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- cos and sec are symmetric across the y-axis. This means they are
even functions, where cos(-x) = cos(x) and sec(-x) = sec(x). The
other trigonometric functions are odd functions, which means
sin(-x) = -sin(x), tan(-x) = -tan(x), csc(-x) = -csc(x), and
cot(-x) = -cot(x).
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Glossary
Acute angle:
angle measuring less than 90 degrees.
Asymptote:
line which the function approaches but never
reaches.
CAST rule:
tells which trigonometric function is positive
in a given quadrant.
Circle:
the set of points which are distance r (called
the radius) from a point c (called the center).
Complementary angles:
a pair of angles whose measures add up to
90 degrees.
Conjugate angles:
a pair of angles whose measures add up to
360 degrees.
Degree:
symbolized °; a unit of measurement equal
to 1/360 of a circle.
Hypotenuse:
the side of a right triangle opposite the right
angle.
Obtuse angle:
angle measuring more than 90 degrees and
less than 180 degrees.
Periodic function:
function for which f(x) = f(x + p) for some
period p.
Pythagorean triple:
a set of three integers a, b and c such that
a2 + b2 = c2.
Pythagorean Theorem:
Given a right triangle with side lengths of a
and b and a hypotenuse of c, a2 + b2 = c2.
Radian:
a unit of angular measure, used with circles.
It is equivalent to 180/π degrees.
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Radius:
the distance between the center of a circle
and a point on the circle.
Ray:
portion of a line starting with a point and
extending to infinity.
Reciprocal:
the reciprocal of x is 1/x, and vice-versa.
Reflex angle:
angle measuring more than 180 degrees and
less than 360 degrees.
Right angle:
a 90° or π/2 radian angle.
Right triangle:
a triangle with a right angle.
Square root:
the square root of x is a number which when
multiplied by itself equals x.
Straight angle:
180 degree angle.
Supplementary angles:
a pair of angles whose measures add up to
180 degrees.
Unit circle:
a circle with radius 1.
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References
http://en.wikipedia.org/
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