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Trigonometry
Purpose
Trigonometry is used to understand the dimensions of triangles. Using the functions and
ratios of trigonometry, the lengths and angles of any triangle can be found. All that is needed
in order to completely solve a triangle are three pieces of information. This information can
be as simple as two of the side lengths and an angle, or three side lengths, or another
combination.
Trigonometry is used by GPS systems in order to verify the distance to your destination as
well as your location in relation to the various landmarks and roads among other thing.
Trigonometric Notation
The two examples of triangles below show the proper labelling scheme. The example on the
left is a general triangle while the example on the right is a right angle triangle.
A
c
X
b
h
y
B
C
a
Y
x
In both triangles we can see that capital letters (A, B, C, X and Y) denote the interior angles
of the triangle, while the corresponding lower case letters (a, b, c, x and y) denote the lengths
of the corresponding opposite sides. The opposite side is the only side that is not touching
the angle you are referring to. For instance angle “A” is opposite to side “a” because they are
not touching in anyway.
The difference between a right triangle and a general triangle is the 90o angle, or right angle,
that all right triangles have. Right triangles also have a hypotenuse, which is always the side
opposite the right angle.
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Right Angle Triangles
Sine, Cosine, and Tangent
We now look at the functions and ratios that trigonometry gives us to help us solve a triangle.
Sine, Cosine and Tangent are trigonometric functions that will allow us to solve for whatever
is needed in our triangle but first we must learn how to properly view the triangle.
The sine, cosine and tangent functions are defined through right angle triangles, so we will
view those first in order to lay a ground work. All of these functions have the three sides of a
right angle triangle in mind. There is the adjacent side, the opposite side and the hypotenuse.
We must first start with an angle of reference, which we will denote by θ (the Greek letter
theta). Imagine that we are standing at this angle in the triangle denoted below by θ.
θ
We can now determine which side is the adjacent, opposite and the hypotenuse allowing us
to use the trigonometric function to solve the triangle. The adjacent side is connected to both
the reference angle and the right angle. In this case the adjacent side is the bottom edge of
the triangle. The opposite side will be the one that the reference angle is opposite to. In this
case the opposite side is the far left edge of the triangle. The hypotenuse is the side that
connects the reference angle to the opposite side. All of the various sides can be seen in the
triangle below when viewed from our reference angle.
Hypotenuse
Opposite
θ
Adjacent
Knowing what each side of the triangle is called, we can now use the trigonometric functions
sine, cosine and tangent to solve for whatever is needed. All of the following operations can
be performed by a scientific calculator.
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Sine
The sine function relies on the reference angle as well as the ratio of the opposite side over
the hypotenuse. The mathematical operation looks like:
Cosine
The cosine function relies on the reference angle as well as the ratio of the adjacent side over
the hypotenuse. The mathematical operation looks like:
Tangent
The tangent function relies on the reference angle as well as the ratio of the opposite side
over the adjacent side. The mathematical operation looks like:
An Easy way to remember
To remember which sides go with which functions there is a single word that may help.
SOH-CAH-TOA
SOH
refers to sine and opposite over hypotenuse.
CAH
refers to cosine and adjacent over hypotenuse.
TOA
refers to tangent and opposite over adjacent.
A second way to remember
For sine:
Sign On Highway.
The S, O and H refer to the sine function and the opposite over hypotenuse ratio respectively.
For cosine:
Cozy At Home.
The C, A, and H refer to the cosine function and the adjacent over hypotenuse ratio
respectively.
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For tangent:
Tan On Arm.
The T, O and A refer to the tangent function and the opposite over adjacent ratio
respectively.
Solving using Sin, Cos and Tan
Find the hypotenuse and the adjacent length.
4m
θ = 30o
Inverse Trig functions
We have talked about the trig functions sine, cosine and tangent and how to solve for
different sides of a triangle. The inverse functions are used in order to find the angles of a
triangle.
To find the inverse trig functions we must solve for angle instead of for the ratio of lengths.
The same ratios are used when discussing trig functions and inverse trig functions. Here is
how the inverse trig functions look.
Inverse sine
Inverse cosine
Inverse tangent
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The inverse trig function can also be represented as arc functions. For instance we can write:
Solving with inverse trig functions
θ2
Find θ 1 and θ 2.
Using θ1 as the reference
angle
3m
Using θ2 as the reference
angle
θ1
5m
General Triangle
If we shift now to regular triangles we have to think about different ways to solve for what is
needed. We can still use the trigonometric functions and inverse trigonometric functions but
we now have to use geometric laws as well.
Sine Law
The Sine Law describes a ratio that hold true for any triangle. The law states that the ratio of
the sine of an angle over its opposite side is equal to the ratio of the sin of a different angle
and its respective opposite side.
The law looks like this mathematically:
Subsequently the Sine Law can also look like:
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Solving a triangle using the sine law
Solve the triangle
A
c=5m
b=6m
B = 30o
C
a
Cosine Law
The cosine law also works for any triangle and is used when only certain information is
given. This law is used when the sine law just will not work. It describes the relationship
between the lengths of the triangle squared minus a cosine relationship. It looks similar to
the Pythagorean Theorem, except with an extra term at the end.
The law looks like this mathematically depending on which side you wish to calculate:
Solving a triangle using the cosine law
Solve for side b.
3m
b
45o
2m
Solve the rest of any triangle using whatever laws or functions necessary for additional
practice.
www.durhamcollege.ca/sals
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/22/2010