Right Angled
Trigonometry
Labeling a Right Triangle
trigonometry, we give each side a
name according to its position in relation
to any given angle in the triangle:
Hypotenuse, Opposite, Adjacent
hypotenuse is
 The _________
always the longest
side of the triangle.
opposite side is
 The _________
the leg directly across
from the angle.
adjacent side is
 The _________
the leg alongside the
angle.
Adjacent
 In

Opposite
Trigonometric Ratios
Adjacent (ADJ)
We define the 3
trigonometric ratios
in terms of fractions
of sides of right
angled triangles.

Opposite (OPP)

SohCahToa
Sine equals Opposite over Hypotenuse
Cosine equals Adjacent over Hypotenuse
Tangent equals Opposite over Adjacent
65
a
Practice Together:
x
Given each triangle,
write the ratio that
could be used to find x
by connecting the
angle and sides given.
32
b
x
YOU DO:
x
c
d
56
Given the triangle,
write all the ratios that
could be used to find x
by connecting the
angle and sides given.
In a right triangle, if we are given
another angle and a side we can find:
 The

third angle of the right triangle:
How?
Using the ‘angle sum of a triangle is 180’
 The

other sides of the right triangle:
How?
Using the trigonometric ratios
Steps to finding the missing sides of a
right triangle using trigonometric ratios:
HYP
61
9.6 cm
ADJ
x
OPP
1.
Redraw the figure
and mark on it HYP,
OPP, ADJ relative
to the given angle
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
HYP
61
9.6 cm
ADJ
x
OPP
Steps to finding the missing sides of a
right triangle using trigonometric ratios:
4.
HYP
61
9.6 cm
ADJ
x
OPP
Solve the equation
to find the
unknown.
7.8 m
xm
41
Practice Together:
Find, to 2 decimal
places, the unknown
length in the triangle.
14.6 m
YOU DO:

bm
am
63
Find, to 1 decimal
place, all the unknown
angles and sides in the
triangle.
Steps to finding the missing angle of a
right triangle using trigonometric ratios:
OPP

2.67 km
HYP
1.
ADJ
Redraw the figure
and mark on it HYP,
OPP, ADJ relative
to the unknown
angle
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
OPP

2.67 km
HYP
ADJ
Steps to finding the missing angle of a
right triangle using trigonometric ratios:
4.
OPP

2.67 km
HYP
ADJ
Solve the equation
to find the unknown
using the inverse of
trigonometric ratio.
Practice Together:

3.1 km
2.1 km
Find, to one decimal
place, the unknown
angle in the triangle.
YOU DO:
4m

7m
Find, to 1 decimal
place, the unknown
angle in the given
triangle.
Practice: Isosceles Triangles
 Using
what we already know about right
angles in isosceles triangles find the
unknown side.
x cm
67
10 cm
YOU DO: Isosceles Triangles
 Find
the unknown angle of the isosceles
triangle using what you already know
about right angles in isosceles triangles.
5.2 m
8.3 m

Practice: Circle Problems
 Use
what you already know about right
angles in circle problems to find the
unknown angle.
6 cm

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
Practice: Other Figures (Trapezoid)
 Find
65
x given:
48
YOU DO: Other Figures (Rhombus)
A
rhombus has diagonals of length 10 cm
and 6 cm respectively. Find the smaller
angle of the rhombus.
