Transcript Trig Ratios

Junior Mathematics – Trig Ratios
Junior Mathematics – Trig Ratios
Here’s a happy student who nailed
this topic!
What are sin cos and tan ??
What are sin cos and tan ??
Clues – used in
Where there are 3 sides.
So, like the sweets, we can combine two
sides in 6 different ways
What are sin cos and tan ??
Clues – used in
They are the 3 ways that two sides can be
selected in relation to one angle
What are sin cos and tan ??
Ɵ
They are the 3 ways that two sides can be
selected in relation to one angle
What are sin cos and tan ??
O (opposite)
Ɵ
They are the 3 ways that two sides can be
selected in relation to one angle
What are sin cos and tan ??
O (opposite)
Ɵ
They are the 3 ways that two sides can be
selected in relation to one angle
What are sin cos and tan ??
H (hypotenuse)
Ɵ
They are the 3 ways that two sides can be
selected in relation to one angle
What are sin cos and tan ??
H (hypotenuse)
O (opposite)
Ɵ
They are the 3 ways that two sides can be
selected in relation to one angle
What are sin cos and tan ??
H (hypotenuse)
O (opposite)
Ɵ
A (adjacent)
between angle and rt L
What are sin cos and tan ??
Ɵ
What are sin cos and tan ??
Ɵ
H (hypotenuse)
What are sin cos and tan ??
Ɵ
H (hypotenuse)
O (opposite)
A (adjacent)
What are sin cos and tan ??
Ɵ
H (hypotenuse)
O (opposite)
So, what are sin cos and tan ??
They are the 3 ratios that two sides make in
relation to one angle
-1
-1
What are sin cos and tan-1 ?
Ɵ
They are the 3 ways that an angle can
be found when given the lengths of two sides
Finding Trig Ratios

A trigonometric ratio is a ratio of the
lengths of two sides of a right triangle.
The word trigonometry is derived from
the ancient Greek language and means
measurement of triangles. The three
basic trigonometric ratios are sine,
cosine, and tangent, which are
abbreviated as sin, cos, and tan
respectively.
Trig Ratios

A
Let ∆ABC be a right
triangle. The sine,
the cosine, and the
tangent of the acute
angle A are
defined as follows.
Side adjacent to A b
cos A =
=
hypotenuse
c
sin A =
Side opposite A
hypotenuse
=
a
c
Side opposite A
a
tan A =
=
Side adjacent to A b
Trig Ratios
A
Hypotenuse
Trig Ratios
A
Hypotenuse
A
The Trig Ratios
SohCahToa
Hypotenuse
cos A
sin A
Side opposite A
=
hypotenuse
=
O
A
tan A
H3 Graphic
Side adjacent to A A
=
=
hypotenuse
H
Side opposite A
O
=
=
Side adjacent to A A
Ex. 1: Finding Trig Ratios

Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
B
17
8
A
15
C
Ex. 1: Finding Trig Ratios

Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
B
H
A
17
15
O
8
C
Ex. 1: Finding Trig Ratios

Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
B
H
17
O
8
A
A
15
C
Ex. 1: Finding Trig Ratios

Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
Soh Cah Toa
o
Soh  Sine A =
h
B
17
8
A
15
C
Ex. 1: Finding Trig Ratios

Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
Soh Cah Toa
B
17
8
A
o 8
Soh  Sine A =
=
h
15
C
Ex. 1: Finding Trig Ratios

Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
Soh Cah Toa
B
17
8
A
o 8
Soh  Sine A =
=
h 17
15
C
Ex. 1: Finding Trig Ratios

Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
Soh Cah Toa
o
Soh  Sine A =
h
B
17
8
A
15
C
Ex. 1: Finding Trig Ratios

Find the sine, the
cosine, and the
tangent ratios for A
in each triangle
beside.
B
17
8
A
15
C
Ex. 1: Finding Trig Ratios
sin A =
cosA =
tanA =
opposite
8≈
17
hypotenuse
adjacent
15 ≈
17
hypotenuse
opposite
8≈
15
adjacent
0.4706
0.8824
0.5333
B
17
8
A
15
C
Trig ratios are often
expressed as decimal
approximations.
Using a Calculator

You can use a calculator to approximate
the sine, cosine, and the tangent of 74.
Make sure that your calculator is in
degree mode. The table shows some
sample keystroke sequences accepted
by most calculators.
Ex 2 – sample keystrokes
Sample keystroke
sequences
SIN
74
ENTER
COS
74
ENTER
TAN
74
ENTER
Sample calculator display
Rounded
To 4 Decimal Places
Ex 2 – sample keystrokes
Sample keystroke
sequences
SIN
Sample calculator display
Rounded
To 4 Decimal Places
0.961262695
0.9613
0.275637355
0.2756
3.487414444
3.4874
74
ENTER
COS
74
ENTER
TAN
74
ENTER
Using Trigonometric Ratios in Real-life

Suppose you stand and look up at a
point in the distance. Maybe you are
looking up at the top of a tree as in
Example 6. The angle that your line of
sight makes with a line drawn
horizontally is called angle of elevation.
Ex. 3: Indirect Measurement

You are measuring the
height of a pine tree in
Alaska. You stand 15m
from the base of the tree.
You measure the angle of
elevation from a point on
the ground to the top of the
top of the tree to be 59°.
To estimate the height of
the tree, you can write a
trigonometric ratio that
involves the height h and
the known length of 15m.
15m
The math
opposite
tan 59°
=
adjacent
h
tan 59°
=
15
15 tan 59° = h
15 (1.6643) = h
24.96m = h
Write the trig ratio
Substitute values
15
Multiply each side by 15
Use a calculator to find tan 59°
Simplify
The tree is about 25m tall
Ex 4: Estimating Distance

Escalators. The escalator
at the Brisbane Central
Rail Station rises 21m at a
30° angle of elevation.

To find the distance d a
person travels on the
escalator stairs, you can
write a trig ratio that
involves the hypotenuse
and the known leg of 21m.
d
30°
21m
Now the maths
opposite
sin 30° =
hypotenuse
Write the ratio
for sine of 30°
21
sin 30° =
d
d sin 30° = 21
21
d=
Multiply each side by d.
Divide each side by sin 30°
sin 30°
21
d=
Substitute values.
0.5
d = 42m
Substitute 0.5 for sin 30°
Simplify
A person travels 42m on the escalator stairs.
Using the inverse Trig ratios to find angles
The angle that an anchor line
makes with the sea bed is really
critical for holding the ship. This
angle depends upon the type of
sea bed, etc. In this case, the sea
bed is mud and the best angle for
holding this ship is between 42-50
degrees.
Will the boat be safely anchored?
Using the inverse Trig ratios to find angles
SohCahToa
The angle that an anchor line
makes with the sea bed is really
critical for holding the ship. This
angle depends upon the type of
sea bed, etc. In this case, the sea
bed is mud and the best angle for
holding this ship is between 42-50
degrees.
Will the boat be safely anchored?
Using the inverse Trig ratios to find angles
SohCahToa
The angle at “a” uses the
opposite side and the
hypotenuse – that is,
“o” and “h”. Therefore, the trig
ratio must also use “o” and
“h”, which is the Sine ratio.
hypotenuse
opposite
The angle that an anchor line
makes with the sea bed is really
critical for holding the ship. This
angle depends upon the type of
sea bed, etc. In this case, the sea
bed is mud and the best angle for
holding this ship is between 42-50
degrees.
Will the boat be safely anchored?
Using the inverse Trig ratios to find angles
opp
Sin a° =
hyp
18.88
Sin a° =
30
Sin a° =
0.8128
sin-1 x sin a° = sin-1 x 0.8128
a
anchor angle, a
= Sin-1 (.8128)
= 54.4 degrees (1dp)
No, the boat will not be safely anchored
The angle that an anchor line
makes with the sea bed is really
critical for holding the ship. This
angle depends upon the type of
sea bed, etc. In this case, the sea
bed is mud and the best angle for
holding this ship is between 42-50
degrees.
Will the boat be safely anchored?