Right Angled Trigonometry Grade 9 Mathematics

Right angles

1. Right Angled
Trigonometry

2. Labeling a Right Triangle
In trigonometry, we give each side a
name according to its position in relation
to any given angle in the triangle:
Hypotenuse, Opposite, Adjacent

H
y
p
o
t
e
n
u
s
e
A
d
ja
c
e
n
t
Opposite
The _________ is
always the longest
side of the triangle.
The _________ side is
the leg directly across
from the angle.
The _________ side is
the leg alongside the
angle.
hypotenuse
opposite
adjacent

3. Trigonometric Ratios
We define the 3
trigonometric ratios
in terms of fractions
of sides of right
angled triangles.

H
y
p
o
t
e
n
u
s
e

(
H
Y
P
)
A
d
ja
c
e
n
t

(
A
D
J
)
Opposite (OPP)

4. SohCahToa
Sine equals Opposite over Hypotenuse
Cosine equals Adjacent over Hypotenuse
Tangent equals Opposite over Adjacent

5. Practice Together:
Given each triangle,
write the ratio that
could be used to find x
by connecting the
angle and sides given.
65
a
x
32
b
x

6. YOU DO:
Given the triangle,
write all the ratios that
could be used to find x
by connecting the
angle and sides given.
56
d
x
c

7. In a right triangle, if we are given
another angle and a side we can find:
The third angle of the right triangle:
How?
The other sides of the right triangle:
How?
Using the ‘angle sum of a triangle is 180’
Using the trigonometric ratios

8. Steps to finding the missing sides of a
right triangle using trigonometric ratios:
1.Redraw the figure
and mark on it HYP,
OPP, ADJ relative
to the given angle
61
9.6 cm
x
HYP
OPP
ADJ

9. Steps to finding the missing sides of a
right triangle using trigonometric ratios:
2.For the given angle
choose the correct
trigonometric ratio
which can be used
to set up an
equation
3.Set up the equation
61
9.6 cm
x
HYP
OPP
ADJ

10. Steps to finding the missing sides of a
right triangle using trigonometric ratios:
4.Solve the equation to
find the unknown.
61
9.6 cm
x
HYP
OPP
ADJ

11. Practice Together:
Find, to 2 decimal
places, the unknown
length in the triangle.
41
x m
7.8 m

12. YOU DO:
Find, to 1 decimal
place, all the unknown
angles and sides in the
triangle.

a m
14.6 m
63
b m

13. Steps to finding the missing angle of a
right triangle using trigonometric ratios:
1.Redraw the figure
and mark on it HYP,
OPP, ADJ relative
to the unknown
angle

5
.
9
2

k
m
HYP
OPP
ADJ
2
.
6
7

k
m

14. Steps to finding the missing angle of a
right triangle using trigonometric ratios:
2.For the unknown
angle choose the
correct trig ratio
which can be used
to set up an
equation
3.Set up the equation

5
.
9
2

k
m
HYP
OPP
ADJ
2
.
6
7

k
m

15. Steps to finding the missing angle of a
right triangle using trigonometric ratios:
4.Solve the equation
to find the unknown
using the inverse of
trigonometric ratio.

5
.
9
2

k
m
HYP
OPP
ADJ
2
.
6
7

k
m

16. Practice Together:
Find, to one decimal
place, the unknown
angle in the triangle.

3.1 km
2.1 km

17. YOU DO:
Find, to 1 decimal
place, the unknown
angle in the given
triangle.

7 m
4 m

18. Practice: Isosceles Triangles
Using what we already know about right
angles in isosceles triangles find the
unknown side.
10 cm
x cm
67

19. YOU DO: Isosceles Triangles
Find the unknown angle of the isosceles
triangle using what you already know
about right angles in isosceles triangles.
8.3 m
5.2 m


20. Practice: Circle Problems
Use what you already know about right
angles in circle problems to find the
unknown angle.
6 cm
1
0

c
m


21. YOU DO: Circle Problems
Use what you already know about right
angles in circle problems to find the
unknown side length.
6.5 cm
56
x c
m

22. Practice: Other Figures (Trapezoid)
Find x given:
1
0
c
m
x

c
m
65 48

23. YOU DO: Other Figures (Rhombus)
A rhombus has diagonals of length 10 cm
and 6 cm respectively. Find the smaller
angle of the rhombus.
10 cm
6

c
m
