Download 17. Trigonometry - Haese Mathematics

Mathematics for Australia Year 9 - Homework
- 2013/1/4 15:04 - page 90
Mathematics for Australia Year 9 - Homework
TRIGONOMETRY (Chapter 17)
CHAPTER 17: TRIGONOMETRY
17A
b Hence estimate the value of:
i sin 34±
ii cos 34±
LABELLING RIGHT ANGLED TRIANGLES
iii tan 34±
REMINDER
c Check your answers using a calculator.
hypotenuse
opposite
µ
adjacent
d Explain the difference between your results in b
and c.
1 For each diagram below, name the:
i hypotenuse
ii side opposite angle µ
iii side adjacent to angle µ.
a A
b
µ
B
C
P
c X
µ
d
Z
Q
2
µ
R
L
µ
M
a Use the diagram alongside
to determine which is
greater:
i sin 55± or sin 70±
55°
ii tan 55± or tan 70±
70°
Y
N
17B
b Check your answers using a calculator.
THE TRIGONOMETRIC RATIOS
3 Consider the triangle
PQR alongside.
a Use Pythagoras’
theorem to find
the unknown
side length.
REMINDER
OPP
.
sin µ =
HYP
ADJ
cos µ =
.
HYP
OPP
tan µ =
.
ADJ
1 Consider the right
angled triangle ABC
alongside.
a Use a ruler to
find the length
of each side,
to the nearest
millimetre.
HYP
P
15.3 mm
R
23°
36.0 mm
Q
Give your answer correct to 1 decimal place.
OPP
µ
ADJ
A
34°
B
b Hence, estimate the value of:
ii cos 23±
i sin 23±
C
91 0
iii tan 23±
Mathematics for Australia Year 9 - Homework
- 2013/1/4 15:04 - page 91
TRIGONOMETRY (Chapter 17)
17C
Mathematics for Australia Year 9 - Homework
19°
f
FINDING SIDE LENGTHS
x cm
REMINDER
Step 1:
On the figure mark HYP, OPP, and ADJ
relative to a given angle.
Step 2:
Choose an appropriate trigonometric ratio,
and construct an equation.
Step 3:
18.5 cm
3 Find, rounded to one
decimal place, all
unknown sides and
angles:
Solve the equation to find the unknown side
length.
25°
4m
bm
am
µ
1 Write down a trigonometric equation connecting the
angle and the sides given:
a
b
x 72°
y
c
30°
b
c
17D
r
55°
FINDING ANGLES
q
REMINDER
2 Find x, giving your answer rounded to 2 decimal places:
a
128 cm
73°
x cm
xm
b
Step 1:
On the figure mark HYP, OPP, and ADJ
relative to the angle you are trying to find.
Step 2:
Choose an appropriate trigonometric ratio,
and construct an equation. (The LHS will be
sin µ, cos µ, or tan µ.)
Step 3:
Use inverse sine, inverse cosine or inverse
tangent on your calculator to find the value
of µ.
47°
27 m
1 Find, to one decimal place, the measure of the angle
marked µ in:
a
10 cm
µ
c
x mm
62°
d
33°
9 cm
93 mm
35 mm
50 mm
b
45 mm
x mm
µ
e
c
3.2 m
40°
4.5 m
xm
91 1
3.6 m
µ
Mathematics for Australia Year 9 - Homework
- 2013/1/4 15:04 - page 92
Mathematics for Australia Year 9 - Homework
d
TRIGONOMETRY (Chapter 17)
In this section, round your answers to 3 significant figures.
2.1 km
µ
1 A telescope is set
1.2 m
up to monitor the
movement of Venus
26°
across the night sky. eye level
The telescope is at
an angle of 26± to
the horizontal, and
the telescope is 1:2 m
long.
How many centimetres above eye level is the other end
of the telescope?
0.8 km
2
44.7 mm
a Find, rounded
to one decimal
place, all
unknown sides
and angles in
the following:
¯
®
xm
46.6 mm
2 A sandwich board is set up
on the footpath. The angle at
the apex is 32± , and the feet
are set 40 cm apart. How
long are the boards that make
up the sign?
b Check your answer for x using Pythagoras’ theorem.
3 Try to find µ in the following
using trigonometry. What
conclusions can you draw?
µ
H
A
E
S
E
32°
8.96 m
40 cm
8.96 m
9m
3 Find the unknown
angles in this
trapezium.
¯
¯
6m
®
17E
®
15 m
PROBLEM SOLVING WITH
TRIGONOMETRY
REMINDER
Use these steps to solve problems involving right angled
triangles:
Step 1:
Step 2:
4 A triangular block
of land is being
measured by a
surveyor.
What
are the values
of the unknown
angles?
Draw a diagram to illustrate the situation.
Mark on the diagram the unknown angle or
side that needs to be calculated. We often use
x for a length and µ for an angle.
Step 3:
Locate a right angled triangle in your
diagram.
Step 4:
Write an equation connecting an angle and
two sides of the triangle using an appropriate
trigonometric ratio.
Step 5:
Solve the equation to find the unknown.
Step 6:
Write your answer in sentence form.
91 2
30 m
18 m
Á
µ
Mathematics for Australia Year 9 - Homework
- 2013/1/4 15:04 - page 93
TRIGONOMETRY (Chapter 17)
Mathematics for Australia Year 9 - Homework
5 Find the perimeter of
the rhombus shown
alongside.
3°
6 An aeroplane, currently at altitude
1500 m, starts its final descent into
the airport. Its angle of descent is
3± . In kilometres, how far away
from the airport runway is the plane
when it begins its descent?
mm
25
20°
1500 m
REVIEW OF CHAPTER 17
1 Find the exact value of:
a sin µ
b cos µ
8
4
µ
c tan µ
5.5 km
3 km
18 m
12°
2 Write a trigonometric equation
connecting the angle and the sides
given:
3 Find the value of ®:
a
®
105 cm
7 A ten pin bowler releases the
ball from the right of his lane.
It travels in a straight line to hit
the 7-pin as shown. What is the
angle µ between the path of the
ball and the side of the lane?
µ
a
b
8 Find the lengths of
the diagonals of this
rectangle.
b
4.55 m
7.6 m
®
10.8 m
P
S
34°
Q
R
9 Determine the height of a boy who casts a 3 metre
shadow when the sun is 20± above the horizon.
4 Find the value of x:
a
x cm
49°
b
21 cm
11 mm
10
a Find the exact value of
x in the triangle given.
®
42 cm
x mm
¯
2 cm
8°
5 Find the remaining side
lengths in the given triangle,
giving your answer correct to
the nearest millimetre:
x cm
A
b Hence, or otherwise, find the unknown angles of the
triangle.
35°
1m
C
B
91 3