7.2 Solving Similar Triangle Problems
Angle of Elevation (inclination): the angle between the horizontal and the line
of sight when looking up at an object
Angle of Depression: the angle between the horizontal and the line of sight
when looking down at an object.
Example 1:
A new bridge is going to be built across a river, but the width of the river cannot be
measured directly. Surveyors set up posts at points A, B, C, D and E. Then they took
measurements relative to the posts. Use the surveyor’s measurements to determine the
width of the river.
Example 2:
Andrea is a landscape designer. She is working on a backyard that is in the shape of a
right triangle. She needs to cover the yard with sod and then fence the yard. She starts
by drawing a scale diagram using the scale 1 cm represents 6.25 m. She marks the
dimensions of the yard on her drawings as 5 cm, 12 cm, and 13 cm. A roll of sod covers
about 0.93 m2. How many rolls of sod does Andrea need? What length of fencing does
she need?
b = 13 cm
c = 12 cm
a = 5 cm
Example 3:
Shiva is standing beside a lighthouse on a sunny day. She measures the length of her
shadow, which is 4.8 m and the length of the shadow cast by the lighthouse, which was
75 m. Shiva is 1.6 m tall. How tall is the lighthouse? (Draw a diagram to help solve the
problem).
7.3 Investigation: The Cosine Ratio
1) Four nested right triangles are drawn on grid paper.
A
B
C
E
D
F
G
H
I
a) Use the Pythagorean Theorem to calculate the lengths of:
AC2 =
AE2=
AG2 =
AI2 =
b) Copy and complete the table. Express each ratio in decimal form.
Triangle
Ratio
ABC
AB
=
AC
AFG
ADE
AD
=
AE
b) How do the ratios compare?
2) Which angle is common to all four triangles?
3a) Name 4 similar triangles.
b) How do you know that the triangles are similar?
AF
=
AG
AHI
AH
=
AI
The ratio you found is called the _____________________ for the common angle.
B
A
Cosine A = _____________________________ or cos A = _______________
Example 1: Find the cosine ratio for C and A in the following triangle.
Express your answer in fraction form.
B
4 cm
3 cm
cos A 
A
C
5 cm
cosC 
The Sine Ratio
1) Four nested right triangles are drawn on grid paper.
A
B
C
D
E
F
G
H
I
a) Use the Pythagorean Theorem to calculate the lengths of:
DE2 =
CE2=
BE2 =
AE2 =
b) Copy and complete the table. Express each ratio in decimal form.
Triangle
Ratio
ECG
EDF
DF
=
DE
CG
=
CE
b) How do the ratios compare?
2) Which angle is common to all four triangles?
3a) Name 4 similar triangles.
EBH
BH
=
BE
EAI
AI
=
AE
b) How do you know that the triangles are similar?
The ratio you found is called the _____________________ for the common angle.
C
B
A
Sine A = _____________________________ or
sin A = _______________
Example 1: Find the sine ratio for C and A in the following triangle.
Express your answer in fraction form.
B
4 cm
3 cm
sin A 
A
C
5 cm
sinC 
The Tangent Ratio
Investigation
1) Four nested right triangles are drawn on grid paper.
H
F
D
B
A
C
E
G
I
a) Copy and complete the table. Express each ratio in decimal form.
Triangle
Ratio
ABC
BC
=
AC
DE
=
AE
b) How do the ratios compare?
2) Which angle is common to all four triangles?
3a) Name 4 similar triangles.
AFG
ADE
FG
=
AG
AHI
HI
=
AI
b) How do you know that the triangles are similar?
The ratio you found is called the _____________________ for the common angle.
C
B
A
tangent A = _____________________________ or tan A = _______________
 The value of the ratio for a given angle depends only on the measure of the
________________
 An _______________ angle of a given measure has a _____________ tangent ratio
 The tangent ratio DOES NOT depend on the size of the ________ angle
Example 1: Find the tangent ratio for C and A in the following triangle.
Express your answer in fraction form.
B
4 cm
3 cm
tan A 
A
C
5 cm
tan C 
7.4 The Primary Trigonometric Ratios
Trigonometry is a branch of mathematics dealing with angles, triangles and
trigonometric functions. The word comes from the Greek trigonon meaning “three
angles” and metro meaning “measure”.
Before we begin, let’s identify the opposite (opp), adjacent (adj) and hypotenuse (hyp)
sides in ABC.
Label the sides relative to A.
Label the sides relative to B
A
A
C
What do you notice?
B
C
B
The sides _______________ and _______________ of a right angled triangle are
labeled according to the angle given or required.
The _______________ is always the longest side or the side opposite of the right angle.
For a given angle A in a right angle triangle, there are three important ratios. These are called:
The primary trigonometric ratios: the sine ratio, the cosine ratio, and the tangent ratio.
We can calculate the ratios of any two of the sides in one triangle using the following:
sin A =
opposite
hypotenuse
cos A =
adjacent
hypotenuse
tan A =
opposite
adjacent
Use the following memory device to help you remember the ratios:
SOH CAH TOA
Example: Find the primary trig ratios for the following triangle. Express as a fraction and
as a decimal correct to four decimal places.
A
5
B
1
2
C
Trigonometric Ratios on a Scientific Calculator
The values of the three primary trigonometric ratios can be found using a scientific
calculator. *Be sure that the calculator is in DEGREE mode for angle calculations.
1. Determine the value of each ratio using your calculator. Round to 4 decimal places.
sin
250 =
a) sin 250 =
b) cos 560 =
c) tan 780 =
2. Determine the size of the angle to the nearest degree.
*press 2nd function or shift before hitting the appropriate trigonometric button to get an
angle measurement.
a) sin A =
3
4
b) cos B = .2397
 A = _________
c) tan C =
 B = _________
 C = _________
3. Find the unknown angle using trigonometry.
A
a)
b) A
1
4
B
9
B
15
•
2
2
2
0
•
C
C
4. Determine the values of tan, sine and cos for angle A and C.
a)
b)
A
A
1
24 2
B
23
67◦
B
C
C
2
3
7.5 Solving Right Triangles
RECALL: sin A =
opposite
hypotenuse
cos A =
adjacent
hypotenuse
tan A =
opposite
adjacent
SOH CAH TOA
Example 1:
A farmers’ co-operative wants to buy and install a grain auger. The auger would be
used to lift grain from the ground to the top of a silo. The greatest angle of elevation
that is possible for the auger is 35○. The auger is 18 m long. Calculate the maximum
height the auger can reach.
18 m
h
35○
Example 2:
55 m
Determine the length of p.
15○
P
Example 3:
Noah is flying a kite and has released 25 m of string. His sister is standing 8 m
away, directly below the kite. What is the angle of elevation of the string?
25 m
8m
Example 4:
In triangle ABC, <A = 90○ , a = 7.8 m, an dc = 5.2 m. Draw the diagram and
solve the missing sides and angles.
opposite
7.6 Solving Right Triangles Problems
SOH CAH TOA
Example 1:
Jackie works for an oil company. She needs to drill a well to an oil deposit. The deposit lies
2300 m below the bottom of a lake, which is 150 m deep. The well must be drilled at an
angle from a site on land. The site is 1000 m away from a point directly above the deposit.
Determine the angle at which the well should be drilled.
Example 2:
Ayesha is a forester. She uses a clinometer (a device used to measure angles of elevation)
to sight the top of a tree. She measures an angle of 48○. She is standing 7.2 m from the
tree, and her eyes are 1.6 m above the ground. How tall is the tree.
Example 3:
A group of students are on an outdoor education trip. They leave their campsite and travel
240 m before reaching the first orientating checkpoint. They turn, creating a 42 ○ angle with
their previous path, and travel another 180 m to get to the second checkpoint. They turn
again and travel the shortest possible path back to their campsite. What area of the woods
did their triangular route cover?
Example 4:
Lyle stood on land, 200 m away from one of the towers on a bridge. He reasoned that he
could calculate the height of the tower by measuring the angle to the top of the tower and
the angle to its base at water level. He measured the angle of elevation to its top as 37 ○
and the angle of depression to its base as 21○. Calculate the height of the tower to the
nearest meter.