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Trigonometry

- Pythagoras Theorem Trigo Ratios of Acute Angles

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

Trigo Ratios of Acute angles

hypotenuse

opposite

adjacent

Hypotenuse side opposite right angle/longest

side

Adjacent side touching theta

Opposite side opposite theta

Trigo Ratios of Acute angles

Hypotenuse AB

Adjacent AC

Opposite BC

Hypotenuse XZ

Adjacent XY

Opposite YZ

Trigo Ratios of Acute angles

hypotenuse

opposite

adjacent

Sine ratio

Cosine ratio

Tangent ratio

tan

cos

sin

Trigo Ratios of Acute angles

hypotenuse

opposite

adjacent

Trigo Ratios of Acute angles

hypotenuse

opposite

adjacent

TOA CAH SOH

Exercise 1

Exercise 1

Exercise 2

Exercise 3

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

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

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

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,

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

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

Further Examples 1

angle BAC?

Let angle BAC be ?.

Further Examples 1

the length of ED?

Further Examples 1

the length of AE?

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 ?.

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 ?.

Exercise 6

In the diagram, angle ADC 30, angle ACB 50,

angle ABD 90 and BC 4 cm. Calculate

(a) angle DAC

Applications Angle of elevation and Angle of

depression

Applications Angle of elevation and Angle of

depression

Example 1

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.

Trigonometric Ratios of Special Angles 30, 45

and 60.

Trigonometric Ratios of Complementary Angles.

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.

Let the height of the cliff TR

Let the distance the boat is from the cliff at

point Q QR

Let the distance travelled by the boat from point

P to point Q PQ

Q2

3 m

?

1.3 m

Let the angle be ?.

Q3

In 15 Secs, distance travelled 140 x 15 2100 m

2100 m

altitude

10

Plane

Let the altitude be a.

Q4

37

65 m

Let the height of cliff be h.

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.

Q6

kite

30 m

h

67

Let the height of kite be h.

Q7

balloon

d

30 m

75

Danny

Let the distance be d.

Q8

23

d

25 m

23

Buoy

Let the distance be d.

Q9

h

60o

50o

x

1000m

Let the height be h.

Q10

Let the height be h.

h

18 m

x

58

46

Q11

h

33o

22o

x

20 m

Let the height be h.

Q12

h

58o

39o

x

A

B

35 m

Let the height be h.

B

Q13(a)

xo

40

xo

E

60

Let the angle of depression be x.

Q13(b)

A

B

40

F

100

D

C

70o

60

E