9 .1 Angles, Rotations, and Degree Measure - - PDF document

Chapter 9: Trigonometry

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SET 1

Chapter 9

Trigonometry

لاُ مـثـثـلات

Chapter 9: Trigonometry

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9.1 Angles, Rotations, and Degree Measure ـياوسلادلاب شايقلاو ، نارودلا ، ارتاـج

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9.2 Radian Measure ـيقلاـطق فـصنلا شاير

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Chapter 9: Trigonometry

8 9.3 Arc Length and Central Angles ـطسكرـملا اـياوسـلا و شوـقلا لوـية

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9.4 Sector Are يرـئادلا عاطـقلا ةحاـسم

There are two main slices of a circle:

 The "pizza" slice which is called a Sector.  And the slice made by a chord which is called a Segment.

The area of a circle = A = π  r2

Where:

A = the area r = the radius d = the diameter

The area of a sector = A (When θ is in radians) The area of a sector = A (When θ is in degrees) Where:

θ = the central angle A = the area of the sector r = the radius

EXAMPLE 9 Find the area of a sector with a central angle of 60 degrees and a radius of 10cm. Express the answer to the nearest tenth. Solution Area of the sector = A

2 2

cm 3 . 52 6 314 100 14 . 3 6 1 10 360 60          EXAMPLE 10 Find the area of a sector with a central angle of 2.16 radians and a radius of 20m. Solution Area of the sector = A

2 2

m 432 400 08 . 1 20 2 16 . 2     

2

2 r   

2

360 r      4

2

d   

2

360 r     

2

2 r   

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9.5 The Trigonometric Ratios ةـيـثـلـثملا بـسنلا

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9.6 The Six Functions Related 9.7 Function Values of 30º, 45º, and 60º

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the next section.

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9.8 Angles of Elevation and Depression ياوزـ افترلئاـفخنلئا و عاـ ضا

Many applications with right triangles involve an angle of elevation or an angle of depression. The angle between the horizontal and a line of sight above the horizontal is called an angle of elevation. The angle between the horizontal and a line of sight below the horizontal is called an angle of

• depression. For example, suppose that you are looking straight ahead and

then you move your eyes up to look at an approaching airplane. The angle that your eyes pass through is an angle of elevation. If the pilot of the plane is looking forward and then looks down, the pilot’s eyes pass through an angle of depression.

EXAMPLE 51 Amna is standing 110 meters from the base of Al-Safa Grocery

• building. She observes that the angle of elevation of the top of the building is 30º.

Find the height of the building. Solution Let h be the height of the building, side adjacent side

• pposite

  tan 110 3 1 110 3 3 110 30 tan h h h     3 110  h

m

110 m

h

30º Al-Safa Grocery Building

Chapter 9: Trigonometry

18 EXAMPLE 56 A man is 180 cm tall and casts a shadow of 3 60

cm long. What is the

angle of elevation of the sun? Solution side adjacent side

• pposite

  tan

3 3 3 3 3 3 3 60 180 tan      

Since tan 600 3 

, angle of elevation of the sun = θ = 600

EXAMPLE 57 A car is seen from a window of a building that is

3 90 feet from the

• ground. If the car is 270 feet away from the building, what is angle of depression of

the car from the building?

Solution side adjacent side

• pposite

  tan

3 3 270 3 90 tan   

Since tan 300

3 3

 , Angle of depression = 03°

cm 180 cm

feet 270 feet

θ

θ