Download Unit 2 Notes – Right Triangle Trigonometry

MFM 2P
UNIT 2 – Right Triangle Trigonometry
THE PYTHAGOREAN RELATION

The triangle is an important shape in construction and architecture
because it is a rigid (stiff – strong) figure.

Right triangles are commonly used to add additional strength to
vertical supports of a building.

There is a special relationship between the area of a square placed
on the longest side of a right triangle and the sum of the areas of
the squares placed on the other two sides.
C
A
Area of Square C
=
Area of Square A
+
Area of Square B
B
MFM 2P
UNIT 2 – Right Triangle Trigonometry
INVESTIGATE:

Calculate the area of each square and state
the relationship between them.
- Square A has side lengths of 4 cm.
- Square B has side lengths of 3 cm
- Square C has side lengths of 5cm.
5 cm
4 cm
C
A
B
3 cm
Area of Square A
Area of Square B
Area of Square C
Area = 4  4
Area = 3  3
Area = 5  5
= 16 cm2
= 9 cm2
= 25 cm2
Relationship:
Area of Square A
16

+
+
Area of Square B
9
=
=
Area of Square C
25
The relationship is known as the Pythagorean Theorem.
MFM 2P
UNIT 2 – Right Triangle Trigonometry
THE PYTHAGOREAN THEOREM
(hypotenuse side)2
(side 1)2 + (side 2)2
=
Side 1
Hypotenuse
Side 2
Example #1:
Calculate the value of c.
c
6 cm
4 cm
Solution:
(hypotenuse)2
=
(side 1)2 +
(side 2)2
(c)2
=
(4)2
+
(6)2
c2
=
16
+
36
c2
=
52
c2
=
52
c
=
7.21 cm
Square root both
sides of the
equation.
MFM 2P
UNIT 2 – Right Triangle Trigonometry
Example #2:
Calculate the value of b.
b
13 m
5m
Solution:
(hypotenuse)2
=
(side 1)2 +
(side 2)2
(13)2
=
(5)2
+
b2
169
=
25
+
b2
169 – 25
=
b2
144
=
b2
144
=
b2
12 m
=
b
MFM 2P
UNIT 2 – Right Triangle Trigonometry
Example #3:
Solution:
Step 1:
A children’s slide 2.9 m long is 1.5 m from the
base of the ladder. How high is the top of the
slide above the ground?
Draw a sketch of the slide and label accordingly.
Slide (2.9 m)
Height Above the Ground (?)
m)
Distance From Base of Ladder (1.5 m)
Step 2:
Use the Pythagorean to determine the height of the slide
above the ground.
(hypotenuse)2
=
(side 1)2 +
(side 2)2
(2.9)2
=
(1.5)2
+
b2
8.41
=
2.25
+
b2
6.16
=
b2
=
b
2.48
Step 3:
Final Statement

The height of the slide above the
ground is 2.5 m.
MFM 2P
UNIT 2 – Right Triangle Trigonometry
LABELLING RIGHT TRIANGLES
Recall:

Angle Measures are Capitalized:
A B C

Side Lengths lower case letters:
abc
A
B
C
AB = c
BC = a
AC = b
MFM 2P
UNIT 2 – Right Triangle Trigonometry
NAMING THE SIDES & ANGLES IN A RIGHT TRIANGLE
 For C in Right ABC:
- b is the Hypotenuse
- c is the Opposite Side of C
- a is the Adjacent Side of C
A
OPPOSITE
HYPOTENUSE
B
C
ADJACENT
 For A in Right ABC:
- b is the Hypotenuse
- a is the Opposite Side of A
- c is the Adjacent Side of A
A
ADJACENT
HYPOTENUSE
B
OPPOSITE
C
MFM 2P
UNIT 2 – Right Triangle Trigonometry
FIND THE RATIO COMPARING SIDE LENGTHS
Given JKL:
Example #1:
a) Write a ratio comparing the length of the side opposite L to the length
of the hypotenuse.
b) Express the ratio as a decimal, rounded to three decimals.
J
12 cm
a)
Opposite = 12
Hypotenuse
16.3
b)
12 ÷ 16.3 = 0.736
16.3 cm
K
11 cm
Example #2:
L
Given DEF write the following ratios:
a) Compare the length of the side adjacent D to the length of the
hypotenuse.
b) Compare the length of the side opposite F to the length of the
adjacent side.
D
7.3 cm
E
a)
8.5 cm
4.4 cm
Adjacent = 7.3
Hypotenuse
8.5
F
b)
Opposite
Adjacent
= 7.3
4.4
MFM 2P
UNIT 2 – Right Triangle Trigonometry
TRIGONOMETRY

The word Trigonometry comes from two Greek terms:
Trigonom

meaning triangle
Metrikos

meaning measure

TRIGONOMETRY is a branch of mathematics that is used to find
the measures of sides and angles in triangles.

Trigonometry is based on the property of similar triangles.
*Recall:
Similar triangles can be used to calculate unknown
heights or widths of objects.
Eg. Determining the height of a mountain or calculating
depth of a canyon.

Who uses trigonometry?
 Surveyors, Architects, Navigators, and Engineers
MFM 2P
UNIT 2 – Right Triangle Trigonometry
THE SINE RATIO
 For any acute angle in a right triangle, the SINE RATIO is:
Sine =
Length of the Side Opposite to the Acute Angle
Length of the Hypotenuse
OR
sin =
opposite
hypotenuse
Example:
D
tan
sinE =
opposite =
hypotenuse
e
f
sinD =
opposite =
hypotenuse
d
f
12
E

10
F
Use the Pythagorean Theorem to determine the hypotenuse (f).
THE SIN KEY ON YOUR CALCULATOR
 Given the degree measure of an angle, the SIN key can be used to find the
sine ratio for the acute angle.
 Given the sine ratio of an angle, the SIN-1 key can be used to find the
degree measure of the angle.
MFM 2P
UNIT 2 – Right Triangle Trigonometry
Example:
Given XYZ.
X
9 cm
Y
13.3 cm
a) Find the sine ratio for Z.
b) Find the measure of Z.
Solution:
a)
Step 1:
Find the length of the hypotenuse, y.
Use the Pythagorean Theorem.
Step 2:
y2
=
92
+
13.32
y2
=
81
+
176.89
y
=
257.89
y
=
16.1 cm
Write the sine ratio.
sin Z =
b)
z
y
=
9
16.1
Use your calculator to determine Z.
Z
=
sin–1
9
16.1

Z = 34
Z
MFM 2P
UNIT 2 – Right Triangle Trigonometry
THE COSINE RATIO
 For any acute angle in a right triangle, the COSINE RATIO is:
Cosine
=
Length of the Side Adjacent to the Acute Angle
Length of the Hypotenuse
OR
cos =
adjacent
hypotenuse
Example:
G
cosG =
adjacent = h
hypotenuse
i
cosH =
adjacent = g
hypotenuse
i
tan
14
H

16
I
Use the Pythagorean Theorem to determine the hypotenuse (i).
THE COS KEY ON YOUR CALCULATOR
 Given the degree measure of an angle, the COS key can be used to find the
cosine ratio for the acute angle.
 Given the cosine ratio of an angle, the COS-1 key can be used to find the
degree measure of the angle.
MFM 2P
UNIT 2 – Right Triangle Trigonometry
Example:
Given XYZ.
Z
3.6 m
X
a) Find the cosine ratio for Y.
b) Find the measure of Y.
5.3 m
Solution:
a)
Step 1:
Find the length of the hypotenuse, x.
Use the Pythagorean Theorem.
Step 2:
x2
=
3.62
+
5.32
x2
=
12.96
+
28.09
x
=
41.05
x
=
6.4 m
Write the cosine ratio.
cos Y =
b)
z
x
=
5.3
6.4
Use your calculator to determine Y.
Y
=
cos –1 5.3
6.4

Y = 34
Y
MFM 2P
UNIT 2 – Right Triangle Trigonometry
SOLVING SIDE LENGTHS
STEPS:
1.
Park your “car” behind the given acute angle.
2.
Determine what side length you must find.
(hypotenuse, opposite, adjacent)
3.
Determine the side length given.
(hypotenuse, opposite, adjacent)
4.
Select a trigonometric ratio (sine/cosine) to use.
5.
Set up a proportion AND solve for the unknown
side length by cross multiplying.
Example #1:
Determine the length of side c.
A
34º
1.
Park your “car” behind angle A.
c
2.
Find adjacent side.
3.
Given hypotenuse.
12 cm
B
4.
Select Cosine ratio (adjacent/hypotenuse).
5.
Set up AND
solve the proportion.

cos 34º
1
=
C
c
12
c
= (cos34º)(12)
c
=
10 cm
MFM 2P
Example #2:
UNIT 2 – Right Triangle Trigonometry
Determine the length of side q.
P
1.
Park your “car” behind angle A.
2.
Find hypotenuse.
Q
14 m
56º
3.
Given opposite side.
R
4.
Select Sine ratio(opposite/hypotenuse).
5.
Set up AND
solve the proportion.

sin 56º
1
=
14
q
(sin56º)(q)
= (14)(1)
(sin56º)(q)
= 14
q
=
14
sin56
q
=
17 m
MFM 2P
UNIT 2 – Right Triangle Trigonometry
THE TANGENT RATIO
 A trigonometric ratio is the ratio of the lengths of two sides in a right
triangle.
 For any acute angle in a right triangle, the TANGENT RATIO is:
Tangent =
Length of the Side Opposite to the Acute Angle
Length of the Side Adjacent to the Acute Angle
OR
tan =
opposite
adjacent
Example:
B
tanA = opposite
adjacent
=
a
b
= 11
13
tanB = opposite
adjacent
=
b
a
=
11
A
13
13
11
C
THE TAN KEY ON YOUR CALCULATOR
 Given the degree measure of an angle, the TAN key can be used to find the
tangent ratio for the acute angle.
 Given the tangent ratio of an angle, the TAN-1 key can be used to find the
degree measure of the angle.
MFM 2P
UNIT 2 – Right Triangle Trigonometry
Example:
Given ABC.
B
14 cm
32 cm
a)
b)
C
Find the tangent ratio for C.
Find the measure of C.
Solution:
a)
Step 1:
Find the length of the adjacent side, a.
Use the Pythagorean Theorem.
a2
Step 2:
+
=
322
a2
=
1024 – 196
a
=
828
a
=
29 cm
Write the tangent ratio.
tan C =
b)
142
c
a
=
14
29
Use your calculator to determine C.
C
=
tan–1 14
29

C = 26
A
MFM 2P
UNIT 2 – Right Triangle Trigonometry
PRACTICE:
a)
Given ABC, determine sine, cosine, and tangent ratios
for C.
A
3 cm
4 cm
B
5 cm
C
sinc =
b)
3
5
4
5
tanc =
In ABC, determine C.
sinc = 3
5
c)
cosc =
= 0.6
** Use SIN-1
C
= 37
In ABC, determine B.
cosB = 3
5
= 0.6
** Use COS-1
B
= 53
= 1.333
** Use TAN-1
B
= 53
OR
tanB = 4
3
3
4
MFM 2P
UNIT 2 – Right Triangle Trigonometry
ANGLE OF DEPRESSION
 The angle of depression is the angle measured downward between the
horizontal and the line of sight from an observer to an object.
OBSERVER
A
OBJECT
 A Is The Angle of Depression
ANGLE OF ELEVATION (INCLINATION)
 The angle of elevation is the angle measured upward between the
horizontal and the line of sight from an observer to an object.
OBJECT
OBSERVER
 B Is The Angle of Elevation
B
Example:
From a point on the gound 30 m from the foot of a tower, the angle of
elevation of the top of the tower is 72. Find the height of the tower, to the
nearest metre.
tan 72 =
(tan 72)(30) =
h
30
h
h
72
30 m
92
= h
MFM 2P
UNIT 2 – Right Triangle Trigonometry