trigonometry

Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.

Trigonometry

?

Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.

Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.Discuss how the following sequence of diagrams allows us to determine the height of the Eiffel Tower without actually having to climb it.

Trigonometry

30o


Trigonometry

35o


Trigonometry

40o


Trigonometry

45o

?What’s he

going to do next?


Trigonometry

45o

324 m

?What’s he

going to do next?

Trigonometry is concerned with the relationship between the angles and sides of triangles. An understanding of these relationships enables unknown angles and sides to be calculated without recourse to direct measurement. Applications include finding heights/distances of objects.

Trigonometry

45o

324 m

324 m

Trigonometry

324 m

Eiffel Tower Facts:

•Designed by Gustave Eiffel.

•Completed in 1889 to celebrate the centenary of the French Revolution.

•Intended to have been dismantled after the 1900 Paris Expo.

•Took 26 months to build.

•The structure is very light and only weighs 7 300 tonnes.

•18 000 pieces, 2½ million rivets.

•1665 steps.

•Some tricky equations had to be solved for its design. 21

( ) tan ( ) ( ) ( )2

H H

x xf x cons tx H x xw x f x dx

The Trigonometric Ratios

A

BC

hypotenuse

opposite

A

B C

hypotenuse

opposite

adjacent

adjacent

Opposite

Sine AHypotenuse

OSinA

H

Adjacent

Cosine AHypotenuse

CosA

AH

Opposite

Tangent AAdjacent

TanO

AA

Make up a Mnemonic!

S O CH A H T O A

The Trigonometric Ratios (Finding an unknown side).

Example 1. In triangle ABC find side CB.

70o

A

BC

12 cmDiagrams

not to scale.

S O H C A H T O A

Opp

07012CB

Sin

0 11.12 70 (13 ) Sin CB pcm d

Example 2. In triangle PQR find side PQ.

22o

P

QR7.2 cm

S O H C A H T O A

0 7.222Cos

PQ 0

7.222

PQCos

1 )7.8 (cmPQ dp

Example 3. In triangle LMN find side MN.

75o

LM

N

4.3 m

S O H C A H T O A

0 4.375Tan

MN 0

4.375

MNTan

1 )1.2 (mMN dp

xo

43.5 m

75 m

Anytime we come across a right-angled triangle containing 2 given sides we can calculate the ratio of the sides then look up (or calculate) the angle that corresponds to this ratio.

Sin 30o = 0.50

Cos 30o = 0.87Tan 30o = 0.58

True Values (2 dp)

S O H C A H T O A

The Trigonometric Ratios (Finding an unknown angle).

0 43.575

0.58Tanx

30o

Example 1. In triangle ABC find angle A.A

BC

12 cm

11.3 cm

Diagrams not to scale.

The Trigonometric Ratios (Finding an unknown angle).

S O H C A H T O A

11.3Sin

12A

0 ( deg )70 Angle A nearest ree

Sin-1(11.3 12) =

Key Sequence

S O H C A H T O A

4.3

1.2Tan N

N (neares7 t degree4 )oAngle

Example 2. In triangle LMN find angle N. L

M

N

4.3 m

1.2 mTan-1(4.3 1.2) =

Key Sequence

S O H C A H T O A

7.2 Q

7.8Cos

23 ( degree)oAngle Q nearest

Example 3. In triangle PQR find angle Q. P

QR7.2 cm

7.8 cm

Cos-1(7.2 7.8) =

Key Sequence

Applications of Trigonometry

A boat sails due East from a Harbour (H), to a marker buoy (B), 15 miles away. At B the boat turns due South and sails for 6.4 miles to a Lighthouse (L). It then returns to harbour. Make a sketch of the trip and calculate the bearing of the harbour from the lighthouse to the nearest degree.

15

6.4

Tan L

HB

L

15 miles

6.4 miles 0Angle 66.9L

360 66.9 293oBearing

SOH CAH TOA

9.5

12

Sin L

12 ft9.5 ft

A 12 ft ladder rests against the side of a house. The top of the ladder is 9.5 ft from the floor. Calculate the angle that the foot of ladder makes with the ground.


Lo

SOH CAH TOA

52oAngle L

Not to Scale

P

570 miles

W

430 miles

Q

An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 430 miles North to a point P before turning left and flying for 570 miles to a second point Q, West of W. It then returns to base.

(a) Make a sketch of the flight.

(b) Find the bearing of Q from P.


SOH CAH TOA

430

Cos P570

0180 41 221Bearing

P 41oAngle

Angles of Elevation and Depression.

An angle of elevation is the angle measured upwards from a horizontal to a fixed point. The angle of depression is the angle measured downwards from a horizontal to a fixed point.

Horizontal25o

Angle of elevation

Horizontal

25oAngle of depression

Explain why the angles of elevation and depression are always equal.

A man stands at a point P, 45 m from the base of a building that is 20 m high. Find the angle of elevation of the top of the building from the man.


20 m

45 m P

SOH CAH TOA

20

P45

Tan

02Angle P (4 deg )nearest ree

A 25 m tall lighthouse sits on a cliff top, 30 m above sea level. A fishing boat is seen 100m from the base of the cliff, (vertically below the lighthouse). Find the angle of depression from the top of the lighthouse to the boat.

100 m

55 m

DC

100

C55

Tan

00 29 D 90 61.2 ( deg )Angle nearest ree

C 61.2oAngle

D

Or more directly since the angles of elevation and depression are equal.

55

D 29100

oTan Angle DSOH CAH TOA

A 22 m tall lighthouse sits on a cliff top, 35 m above sea level. The angle of depression of a fishing boat is measured from the top of the lighthouse as 30o. How far is the fishing boat from the base of the cliff?

x m

57 m

30o

6057x

Tan

57

30Tanx

60o

99m

57 60

= (nearest m)

x Tan

57

9930

x mTan

Or more directly since the angles of elevation and depression are equal.

30o

SOH CAH TOA

trigonometry

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