Download Day-86-Presentation-Introduction to trigonometric ratios

DAY 86 – INTRODUCTION TO
TRIGONOMETRIC RATIOS
INTRODUCTION
The word trigonometry is coined from two Greek
words, trigonon meaning ‘triangle’ and metron
meaning ‘measure’. Trigonometry is a branch of
mathematics that explores the relationships
between the sides and angles of triangles and the
related calculations. Trigonometry is applied in
several fields which deal with measurements such
as engineering, architecture, surveying, navigation
among others to find unknown angles and lengths.
In this lesson, we will study ratios of the lengths
of sides of a right triangle with reference to its acute
angles and define trigonometric ratios of angles.
VOCABULARY
Trigonometric ratio
The ratio of the length of any two sides of a right
triangle with respect to a related angle.
SIDES AND ANGLES OF A RIGHT TRIANGLE
A right triangle has one right angle and two acute
angles.
Each of its three sides is given a special name.
Its longest side is always opposite the right angle
and it is referred to as the hypotenuse.
The other two sides, namely the opposite sides
and the adjacent sides are named with respect to
the acute angle under consideration in the triangle.
The adjacent side and the opposite side change
depending on the reference angle.
Every acute angle is between two sides, the
hypotenuse and one of the legs.
NAMING THE SIDES OF A RIGHT TRIANGLE
In trigonometry, we sometimes name the sides of a
triangle using a lower case letter opposite the angle
identified by a single capital letter. This is
illustrated below. Consider right ∆ABC below.
A
𝛼
𝑏
𝑐
𝜃
B
𝑎
C
There are two acute angles in ∆ABC, 𝛼 and 𝜃.
With reference to 𝛼:
AC = 𝑏 is the hypotenuse.
AB = 𝑐 is the adjacent side, the side next to 𝛼
BC = 𝑎 is the opposite side, the side opposite to 𝛼
With reference to 𝜃:
AC = 𝑏 is the hypotenuse.
BC = 𝑎 is the adjacent side, the side next to 𝜃
AB = 𝑐 is the opposite side, the side opposite to 𝜃
BASIC TRIGONOMETRIC RATIOS
The ratios of the side of a right triangle are
generally referred to as trigonometric ratios.
The three basic trigonometric ratios are the sine,
shortened to sin, the cosine, shortened to cos and
the tangent, shortened to tan.
Let us define these ratios using right ΔPQR below.
𝛼 is the reference angle.
Note that the names opposite, adjacent, and
hypotenuse in the ratios refer to the lengths of
sides.
It is important to note that, for a given angle 𝛼, the
ratios remain constant however large or small the
triangle is, because the triangles will be similar.
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
P
Q
𝛼
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
R
The trigonometric ratios are defined using the sides
as shown below:
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑃𝑄
𝑆𝑖𝑛 𝛼 =
=
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑃𝑅
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑄𝑅
𝐶𝑜𝑠 𝛼 =
=
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑃𝑅
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑃𝑄
𝑇𝑎𝑛 𝛼 =
=
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑄𝑅
From the definitions above, it is evident that:
Sine is ‘Opposite over Hypotenuse’
Cosine is ‘Adjacent over Hypotenuse’ and
Tangent is ‘Opposite over Adjacent’
We can easily remember this ratios using the
acronym SOH CAH TOA.
We can also obtain other three trigonometric ratios
from the sine, the cosine and the tangent.
These ratios are the cosecant, shortened to cosec
or sometimes csc, the secant, shortened to sec
and the cotangent, shortened to cot.
Let us define these ratios using right ΔPQR below.
Let us define these ratios using right ΔPQR below.
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
P
Q
𝛼
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
R
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑃𝑅
1
𝑐𝑠𝑐 𝛼 =
=
=
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑃𝑄 sin 𝛼
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑃𝑅
1
𝑠𝑒𝑐 𝛼 =
=
=
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑄𝑅 cos 𝛼
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑄𝑅
1
𝑐𝑜𝑡 𝛼 =
=
=
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑃𝑄 tan 𝛼
Example
Using 𝛽 as the reference angle in right ΔKLM below,
identify:
K
𝛽
L
(a) the hypotenuse
M
(b) the opposite side
(c) the adjacent side
Solution
(a) The hypotenuse is the side opposite the right
angle, therefore, KM is the hypotenuse
(b) The opposite side is the opposite 𝛽, LM is the
opposite side.
(c) The adjacent side is next to 𝛽, KL is the
adjacent side.
HOMEWORK
Name the hypotenuse, the adjacent side and the
opposite side with respect to angle 𝜔.
P
𝜔
M
N
ANSWERS TO
MP is the hypotenuse
NP is the adjacent side
MN is the opposite side
HOMEWORK
THE END