Download 2 - Schools

Introduction to
Trigonometry
Right Triangle Trigonometry
Introduction
What special theorem do you already
know that applies to a right triangle?
 Pythagorean Theorem:
a2 + b2 = c2

c
a
b
Introduction
Trigonometry is a branch of mathematics
that uses right triangles to help you solve
problems.
 Trig is useful to surveyors, engineers,
navigators, and machinists (and others
too.)

A trigonometric ratio is a ratio of the lengths of two sides
of a right triangle.
The three basic trigonometric ratios are sine, cosine, and
tangent, which are abbreviated as sin, cos, and tan,
respectively.
When a right triangle has an acute angle with a certain
measure, it is similar to all other right triangles with that
same size acute angle.
That means that the ratios of the sides for any right
triangle with the same sized acute angle are equal,
regardless of the size of the triangle.
sin θ, cos θ, and tan θ will have the same value for these
two triangles because the angle θ is the same in both.
θ
θ
Before we can understand the trigonometric ratios we need
to know how to label Right Triangles.
Labeling Right Triangles
The most important skill you need right
now is the ability to correctly label the
sides of a right triangle.
 The names of the sides are:

 the
hypotenuse
 the opposite side
 the adjacent side
Labeling Right Triangles

The hypotenuse is easy to locate because
it is always found across from the right
angle.
Since this side is
across from the right
angle, this must be
the hypotenuse.
Here is the
right angle...
Labeling Right Triangles
Before you label the other two sides you
must have a reference angle selected.
 It can be either of the two acute angles.
 In the triangle below, let’s pick angle B as
This will be our
the reference angle. B
reference angle...

A
C
Labeling Right Triangles
Remember, angle B is our reference
angle.
 The hypotenuse is side BC because it is
across from the right angle.

B (ref. angle)
hypotenuse
A
C
Labeling Right Triangles

Side AC is across from our reference
angle B. So it is labeled: opposite.
B (ref. angle)
hypotenuse
A
C
opposite
Labeling Right Triangles
Adjacent means beside or next to

The only side unnamed is side AB. This
must be the adjacent side.
B (ref. angle)
adjacent
hypotenuse
C
A
opposite
Labeling Right Triangles
Let’s put it all together.
 Given that angle B is the reference angle,
here is how you must label the triangle:

B (ref. angle)
hypotenuse
adjacent
C
A
opposite
Labeling Right Triangles
Given the same triangle, how would the
sides be labeled if angle C were the
reference angle?
 Will there be any difference?

Labeling Right Triangles
Angle C is now the reference angle.
 Side BC is still the hypotenuse since it is
across from the right angle.

B
hypotenuse
A
C (ref. angle)
Labeling Right Triangles

However, side AB is now the side
opposite since it is across from angle C.
B
opposite
hypotenuse
A
C (ref. angle)
Labeling Right Triangles

That leaves side AC to be labeled as
the adjacent side.
B
hypotenuse
opposite
A
C (ref. angle)
adjacent
Labeling Right Triangles
Let’s put it all together.
 Given that angle C is the reference
angle, here is how you must label the
triangle:

B
hypotenuse
opposite
C (ref. angle)
A
adjacent
Finding Trigonometric Ratios
A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.
TRIGONOMETRIC RATIOS
Let ABC be a right triangle. The sine, the cosine, and the tangent of the
acute angle A are defined as follows.
o
side opposite A
=
sin A =
h
hypotenuse
cos A =
side adjacent A
a
=
hypotenuse
h
o
side opposite A
tan A =
=
a
side adjacent to A
B
hypotenuse
h
A
side
o opposite
A
C
a
side adjacent to A
The value of the trigonometric ratio depends only on the measure of the acute
angle, not on the particular right triangle that is used to compute the value.
How do I remember these?
opposite
hypotenuse
adjacent
Finding Trigonometric Ratios
Find the sine, the cosine, and the tangent of the indicated angle.
R
S
13
5
SOLUTION
T
S
12
The length of the hypotenuse is 13. For S, the length of the opposite side
is 5, and the length of the adjacent side is 12.
sin S =
cos S =
opp.
5
=
hyp.
13
 0.3846
adj.
12
 0.9231
=
hyp.
13
opp.
5
 0.4167
tan S =
=
adj.
12
R
13
5
opp.
T
hyp.
12 adj.
S
Finding Trigonometric Ratios
Find the sine, the cosine, and the tangent of the indicated angle.
R
R
13
5
SOLUTION
T
S
12
The length of the hypotenuse is 13. For R, the length of the opposite side
is 12, and the length of the adjacent side is 5.
sin R =
cos R =
opp. 12
=
hyp.
13
 0.9231
adj.
5
=
 0.3846
hyp.
13
opp.
12
tan R =
=
= 2.4
adj.
5
R
13
5
adj.
T
hyp.
12 opp.
S