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## Trigonometry

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### Trigonometry Pythagoras Theorem & Trigo Ratios of Acute Angles Pythagoras Theorem a2 + b2 = c2 Trigo Ratios of Acute angles Trigo Ratios of Acute angles Trigo Ratios ... – PowerPoint PPT presentation

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Title: Trigonometry

1
Trigonometry
• Pythagoras Theorem Trigo Ratios of Acute Angles

2
Pythagoras Theorem
• a2 b2 c2

where c is the hypotenuse while a and b are the
lengths of the other two sides.
c
a
b
3
Trigo Ratios of Acute angles
hypotenuse
opposite
Hypotenuse side opposite right angle/longest
side
Opposite side opposite theta
4
Trigo Ratios of Acute angles
Hypotenuse AB
Opposite BC
Hypotenuse XZ
Opposite YZ
5
Trigo Ratios of Acute angles
hypotenuse
opposite
Sine ratio
Cosine ratio
Tangent ratio
tan
cos
sin
6
Trigo Ratios of Acute angles
hypotenuse
opposite
7
Trigo Ratios of Acute angles
hypotenuse
opposite
TOA CAH SOH
8
Exercise 1
9
Exercise 1
10
Exercise 2
11
Exercise 3
12
Exercise 4
sin (30 ? 2) 0.2588 sin 30 ? 2 0.25
sin (? ? 2) sin ? ? 2
FALSE
cos 2? 2 ? cos ?
cos (2 30) 0.5 2 cos 30 1.732
FALSE
tan (10 30) 0.839 tan 10 tan 30
0.753
FALSE
tan (A B) tan A tan B
13
Exercise 5
sin ? 0.4537
? sin-1 0.4537 26.98127.0
cos ? 0.3625
? cos-1 0.3625 68.74668.7
tan ? 4.393
? tan-1 4.393 77.17677.2
14
Exercise 5
sin ? 0.8888
? sin-1 0.8888 62.72262.7
cos ? 0.9999
? cos-1 0.9999 0.81020.8
tan ? 0.5177
? tan-1 0.5177 27.37027.4
15
Further Examples 1
In the diagram, BCE is a straight line, angle
ECD 54.8 and angle CDE angle ACB 90. BC
7 cm and AC CE 8 cm. Calculate
angle CED, angle DCB, angle BAC, the length of
ED, the length of AE,
16
Further Examples 1
In the diagram, BCE is a straight line, angle
ECD 54.8 and angle CDE angle ACB 90. BC
7 cm and AC CE 8 cm. Calculate
angle CED?
angle CED 180 - 90 - 54.8 35.2
17
Further Examples 1
In the diagram, BCE is a straight line, angle
ECD 54.8 and angle CDE angle ACB 90. BC
7 cm and AC CE 8 cm. Calculate
angle DCB?
angle DCB 180 - 54.8 125.2
18
Further Examples 1
angle BAC?
Let angle BAC be ?.
19
Further Examples 1
the length of ED?
20
Further Examples 1
the length of AE?
21
Further Examples 2
A 16 m ladder is leaning against a house. It
touches the bottom of a window that is 12 m above
the ground. What is the measure of the angle
that the ladder forms with the ground?
Let the angle be ?.
22
Further Examples 3
A 16 m ladder is leaning against a house. It
touches the bottom of a window that is 12 m above
the ground. What is the measure of the angle
that the ladder forms with the ground?
Let the angle be ?.
23
Exercise 6
In the diagram, angle ADC 30, angle ACB 50,
angle ABD 90 and BC 4 cm. Calculate
(a) angle DAC
24
Applications Angle of elevation and Angle of
depression
25
Applications Angle of elevation and Angle of
depression
26
Example 1
27
Example 2
A surveyor is 100 meters from the base of a dam.
The angle of elevation to the top of the dam
measures
. The surveyor's eye-level is 1.73 meters above
the
ground. Find the height of the dam.
28
Trigonometric Ratios of Special Angles 30, 45
and 60.
29
Trigonometric Ratios of Complementary Angles.
30
At the point P, a boat observes that the angle of
elevation of the cliff at point T is 32o, and the
distance PT is 150m. It sails for a certain
distance to reach point Q, and observes that the
angle of elevation of the point T becomes 48o.
(i) Calculate the height of the cliff. (ii)
Calculate the distance the boat is from the
cliff at point Q. (iii) Calculate the distance
travelled by the boat from point P to point Q.
31
Let the height of the cliff TR
32
Let the distance the boat is from the cliff at
point Q QR
33
Let the distance travelled by the boat from point
P to point Q PQ
34
Q2
3 m
?
1.3 m
Let the angle be ?.
35
Q3
In 15 Secs, distance travelled 140 x 15 2100 m
2100 m
altitude
10
Plane
Let the altitude be a.
36
Q4
37
65 m
Let the height of cliff be h.
37
Q5
Let the height of cliff be h.
tower
cliff
65
53
60 m
Let the height of cliff and tower be x.
Let the height of tower be t.
38
Q6
kite
30 m
h
67
Let the height of kite be h.
39
Q7
balloon
d
30 m
75
Danny
Let the distance be d.
40
Q8
23
d
25 m
23
Buoy
Let the distance be d.
41
Q9
h
60o
50o
x
1000m
Let the height be h.
42
Q10
Let the height be h.
h
18 m
x
58
46
43
Q11
h
33o
22o
x
20 m
Let the height be h.
44
Q12
h
58o
39o
x
A
B
35 m
Let the height be h.
45
B
Q13(a)
xo
40
xo
E
60
Let the angle of depression be x.
46
Q13(b)

A
B
40
F
100
D
C
70o
60
E