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

SET 1 Chapter 9 Trigonometry لاُ مـثـثـلات Chapter 9: Trigonometry 1

9 .1 Angles, Rotations, and Degree Measure ـياوسلادلاب شايقلاو ، نارودلا ، ارتاـج Chapter 9: Trigonometry 2

Chapter 9: Trigonometry 3

Chapter 9: Trigonometry 4

9 .2 Radian Measure ـيقلاـطق فـصنلا شاير Chapter 9: Trigonometry 5

Chapter 9: Trigonometry 6

Chapter 9: Trigonometry 7

9 .3 Arc Length and Central Angles ـطسكرـملا اـياوسـلا و شوـقلا لوـية Chapter 9: Trigonometry 8

Chapter 9: Trigonometry 9

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 .    2 d The area of a circle = A = π  r 2 4 Where: A = the area r = the radius d = the diameter    (When θ is in radians) 2 The area of a sector = A r 2      2 (When θ is in degrees) The area of a sector = A r 360 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.      2 r Solution Area of the sector = A 360 60     2 10 360 1    3 . 14 100 6 314  6  2 52 . 3 cm EXAMPLE 10 Find the area of a sector with a central angle of 2.16 radians and a radius of 20m.    2 Solution Area of the sector = A r 2 2 . 16   2 20 2   1 . 08 400  2 432 m Chapter 9: Trigonometry 10

9 .5 The Trigonometric Ratios ةـيـثـلـثملا بـسنلا Chapter 9: Trigonometry 11

Chapter 9: Trigonometry 12

Chapter 9: Trigonometry 13

9 .6 The Six Functions Related 9 .7 Function Values of 30º, 45º, and 60º Chapter 9: Trigonometry 14

Chapter 9: Trigonometry 15

the next section. Chapter 9: Trigonometry 16

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 . Al-Safa h Grocery Building 30º 110 m Solution Let h be the height of the building, opposite side   tan adjacent side h   tan 30 110 3 h  3 110 1 h  3 110 110  m h 3 Chapter 9: Trigonometry 17

EXAMPLE 5 6 A man is 180 cm tall and casts a shadow of 60 3 cm long. What is the angle of elevation of the sun? 180 cm cm opposite side   Solution tan adjacent side 180   tan 60 3 3  3  3 3   3 3  Since tan 60 0 , angle of elevation of the sun = θ = 60 0 3 EXAMPLE 5 7 A car is seen from a window of a building that is 90 3 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? θ feet θ 270 feet opposite side   Solution tan adjacent side 90 3   tan 270 3  3 3  Since tan 30 0 , 3 Angle of depression = 03 ° Chapter 9: Trigonometry 18