Best Practice #1
Trigonometry
AusVELS Level 9.0
Students will identify similar triangles if the
corresponding sides are in ratio and all corresponding
angles equal.
Measurment and Geometry
Similar Triangles
Similarity can be used to find lengths or heights of
large objects. It leads into problems involving
trigonometry
Measurment and Geometry
Best Practice #2
Trigonometry
AusVELS Level 9.0
Students will identify the hypotenuse, adjacent and
opposite sides with respect to the reference angle in a
right-angled triangle
Measurment and Geometry
Labelling the right-angled triangle
Trigonometry is concerned with the
connection between the sides and
angles in any right angled triangle.
Angle
Measurment and Geometry
The sides of a right -angled triangle are
given special names:
The hypotenuse, the opposite and the
adjacent.
The hypotenuse is the longest side and is
always opposite the right angle.
The opposite and adjacent sides refer to
the reference angle, other than the 90o.
A
A
Measurment and Geometry
Best Practice #3
Trigonometry
AusVELS Level 9.0
Students will define the Sine, Cosine and Tangent ratios
for angles in right-angled triangles
Students will use Trigonometric notation eg. Sin A
Measurment and Geometry
Trigonometric Notation
There are three formulae
involved in trigonometry:
Sin θ=
Cos θ=
Tan θ =
Measurment and Geometry
S OH C AH T OA
Measurment and Geometry
Similar Triangles and Trig Ratios
ABC  QPR
R
B
5
3
C
4
20
12
A
P
Let’s look at the 3 basic Trig
ratios for these 2 triangles
They are similar triangles,
since ratios of
corresponding sides are
the same. All similar
triangles have the same trig
ratios for corresponding
angles.
Q
16
Sin Q

12
20
4
5
Cos Q

16
20
3
4
Tan Q

12
16
Sin A
3

5
Cos A

Tan A

Measurment and Geometry
Best Practice #4
Trigonometry
AusVELS Level 9.0
Students will select the appropriate trigonometric ratio to
calculate unknown sides in a right-angled triangle,
including using trigonometry to find an unknown side
when the unknown letter is on the bottom of the fraction
Measurment and Geometry
Calculating an Unknown Side
To find a missing side from a right-angled
triangle we need to know one angle and one
other side.
Note: If
x
Cos45 = 13
To leave x on its own we need to
move the ÷ 13.
It becomes a “times” when it moves.
x = 13 x Cos(45)
Measurment and Geometry
1.
H
7 cm
k
A
30o
We have been given
the adj and hyp so we
use COSINE:
Cos A =
adjacent
hypotenuse
Cos A = a
h
Cos 30 = k
7
k = 7 x Cos 30
k = 6.1 cm
Measurment and Geometry
2.
We have been given
the opp and adj so we
use TAN:
50o
4 cm
A
Tan A =
r
O
Tan A = o
a
r
Tan 50 =
4
r = 4 x Tan 50
r = 4.8 cm
Measurment and Geometry
3.
k
O
We have been given
the opp and hyp so we
use SINE:
H
12 cm
Sin A =
25o
o
h
sin 25 = k
12
sin A =
k = 12 x Sin 25
k = 5.1 cm
Measurment and Geometry
There are occasions when the unknown
letter is on the bottom of the fraction after
substituting.
Cos45 = 13
u
Move the u term to the other side.
It becomes a “times” when it moves.
Cos45 x u = 13
To leave u on its own, move the cos 45
to other side, it becomes a divide.
u =
13
Cos 45
Measurment and Geometry
Best Practice #5
Trigonometry
AusVELS Level 9.0
To be able to use trigonometry to find an unknown side
when the unknown letter is on the bottom of the fraction.
Measurment and Geometry
Calculating an Unknown Side- pronumeral
on the bottom
When the unknown letter is on the bottom
of the fraction we can simply swap it with
the trig (sin A, cos A, or tan A) value.
Cos45 = 13
u
u =
13
Cos 45
Measurment and Geometry
Cos A = A
1.
x
H
H
Cos 30 = 5
x
5
x =
cos 30
30o
5 cm
A
x = 5.8 cm
sin A = O
2.
8 cm
O
H
m
25o
H
sin 25 = 8
m
8
m=
sin25
m = 18.9 cm
Measurment and Geometry
Best Practice #6
Trigonometry
AusVELS Level 9.0
Students will select the appropriate trigonometric ratio to
calculate unknown angles in a right-angled triangle
Measurment and Geometry
Calculating an Unknown Angle
To do this on the calculator, we use the
sin-1, cos-1 and tan-1 function keys.
Measurment and Geometry
Example:
1. sin θ = 0.1115 find angle θ.
sin-1
(
shift
0.1115
=
)
sin
θ = sin-1 (0.1115)
θ = 6.4o
2.
cos θ = 0.8988 find angle θ
cos-1
(
shift
0.8988
cos
=
)
θ = cos-1 (0.8988)
θ = 26° (to the nearest degree)
Measurment and Geometry
Finding an angle from a triangle
To find a missing angle from a right-angled
triangle we need to know two of the sides of
the triangle.
We can then choose the appropriate ratio,
sin, cos or tan and use the calculator to
identify the angle from the decimal value of
the ratio.
1.
Find angle θ
14 cm
6 cm
θ
a) Identify/label the
names of the sides.
b) Choose the ratio that
contains BOTH of the
letters.
Measurment and Geometry
1.
H
14 cm
6 cm
A
θ
We have been given
the adjacent and
hypotenuse so we use
COSINE:
Cos θ =
adjacent
hypotenuse
Cos θ = A
H
Cos θ = 6
14
Cos θ = 0.4286
θ = cos-1 (0.4286)
θ = 64.6o
Measurment and Geometry
2. Find angle x
3 cm
A
Given adj and opp
need to use tan:
θ
opposite
Tan θ = adjacent
8 cm
O
Tan θ = o
a
Tan θ = 8
3
Tan θ = 2.6667
θ = tan-1 (2.6667)
θ = 69.4o
Measurment and Geometry
3.
10 cm
12 cm
Given opp and hyp
need to use sin:
opposite
Sin θ = hypotenuse
θ
o
h
sin θ = 10
12
sin θ =
sin θ = 0.8333
θ = sin-1 (0.8333)
θ = 56.4o
Measurment and Geometry