SUBJECT: Mathematics
GRADE: 10
CHAPTER / MODULE: 3
UNIT / LESSON TOPIC: trigonometry problems in two dimensions in
right-angled triangles
triangles
UNIT OUTCOMES
LO 3 AS 10.3.5 ; 10.3.6
By the end of this unit, you should be able to :

solve problems in two dimensions by using the trigonometric
functions in right- angled triangles by constructing and
interpreting geometric and trigonometric models including
scale drawings, maps and building plans.
LESSON OVERVIEW: (Knowledge areas)
INTRODUCTION
1. Practice solving right-angled triangles by changing the direction of
the triangle so that the side opposite and the side adjacent are swopped
around. The hypotenuse is always opposite the right angle. – 30 minutes
2. Examples of a right-angled triangle with students and explained
thoroughly – 30 minutes
LESSON:
sin
o
h
cos
a
h
tan
o
a
o = opposite
a = adjacent
h = hypotenuse
1
The following diagram can be shown for the first table :-
r
y
x
The Cartesian plane has four quadrants :*From 0 to 90 is the first quadrant : all three ratios are positive
in this quadrant.
* From 90 to 180 is the second quadrant : sin is positive, cos and tan are
negative in this quadrant.
* From 180 to 270 is the third quadrant : tan is positive, sin and cos are
negative in this quadrant.
* From 270 to 360 is the fourth quadrant : cos is positive, sin and tan are
negative in this quadrant.
ANGLE OF ELEVATION
An angle of elevation is an angle starting from the base line and moving in an
upward direction .
2
ANGLE OF DEPRESSION
An angle of depression is an angle starting from the base line and moving in a
downward direction.
RIGHT-ANGLED TRIANGLES
When we need to solve for right-angled triangles we must be given either :* any two sides of the right-angled triangle or
* one side and the size of an acute angle.
Eg 1.
C
If B is the base
angle of the triangle
then 1 becomes the
opposite side, 2
becomes the
hypotenuse and 3
becomes the
adjacent side.
If C is the base
angle, then 1
becomes the
adjacent side and 3
becomes the
opposite side.
2
1
4
A
3
B
3
1. Find angle A :-
If A is the base
angle then 190m is
the opposite side to
the angle, 84m is
the adjacent side to
the angle and the
side opposite the
right angle is the
hypotenuse.
190m
84m
A
SOLUTION
190
 tan A
84
1. tan A  2,261904762
( shift tan 1 )
A  66,15
Before starting to answer the question, always determine the three sides :Opposite
Adjacent
Hypotenuse
Then use the table above showing the three ratios
sin
o
h
cos
a
h
tan
o
a
2. A fireman’s ladder is inclined at an angle of 58 . The ladder is 10 meters long
measured from the top rung, where the fireman is standing. How high is the
fireman above the ground ?
4
SOLUTION
2.
If a story problem is given, always draw the right-angled triangle to make the
problem easier to solve.
h
10m
58
h
 sin 58
10
h  10  sin 58
h  8,5m
h represents the height of the top rung .
The answer has the  (approximate) sign because the solution has been
rounded off .
REMEMBER

If looking for the angle ( eg. sin
must be used.

If looking for the number ( eg. sin 30 ) use the sin , cos or
tan button without pressing the shift button.
5
= 0,5 ) the “ shift “ button
HOMEWORK
1. Calculate the length of FG.
G
H
3
F
2. Calculate the size of
to one decimal place.
P
5
Q
8
R
3. The angle of depression of a boat from the top of a cliff is 55 .
The boat is 70 meters from the foot of the cliff.
3.1.
What is the angle of elevation of the top of the cliff from the boat ?
3.2.
Calculate the height of the cliff.
6