6.TRIGONOMETRY

1. 6.TRIGONOMETRY

2. CONTENTS • Angle Measurement : Degree and Radian • Trigonometric Ratios in Right-Angled Triangles • Trigonometric Ratio of Special Angles • Simple Trigonometric Equations

3. Plotting Graph of Trigonometric Function • Polar Coordinates • Relationships Among Trigonometric Ratios of an Angles • Sine Rule, Cosine Rule and the Area of Triangles • Angles of Elevation and Angle of Depression

4. TRIANGLES • Right-angled triangle (segitiga siku-siku) • Isosceles triangle (segitiga sama kaki) • Equilateral triangle (segitiga sama sisi) The sum of angles in a triangle is always

5. 1radian A O A.ANGLE MEASUREMENT B • The measurement of = 1 radian radians = radians = radians 1 radian =

6. EXAMPLE : • Convert to radians : a. b. c. • Convert the following radian measure to degrees : a. b. c.

7. hypotenuse C A B adjacent hypotenuse b opposite a A B c adjacent B. Trigonometric Ratios in Right-angled Triangles Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant of an Angle C Figure 6.2 opposite

8. Definition of On a right-angled triangle, the sine of an angle is the ratio between the side opposite to the angle and the hypotenuse. In Figure 6.2 :

9. Definition of On a right-angled triangle, the cosine of an angle is the ratio between the side adjacent to the angle and the hypotenuse. In figure 6.2 :

10. Definition of On a right-angled triangle, the tangent of an angle is the ratio between the side opposite to the angle and the side adjacent to the angle. In figure 6.2 :

11. Definition of On a right-angled triangle, the cosecant of an angle is the ratio between the hypotenuse with the right-angled side opposite to the angle. In figure 6.2 :

12. Definiton of On a right-angled triangle, the secant of an angle is the ratio between the hypotenuse with the right-angled side adjacent to the angle. In figure 6.2 :

13. Definition of On a right-angled triangle, the cotangent of an angle is the ratio between the side adjacent to the angle with the side opposite to the angle. In figure 6.2 :

14. EXAMPLE : • If , determine the value of the • other trigonometric ratios • 2. Given with a right angle at B. If AB = 1 cm and AC = 3 cm, calculate all trigonometric ratios .

15. y P(x,y) r y x x O N C. Trigonometric Ratios of Special Angles

16. y x a. If

17. y x b. If

18. a a/2 a c. If

20. a a d. If

21. Example : 1. If and , determine the length of c and b (in right-angle triangles) 2. Calculate the value of :

22. Y II I X O III IV Figure 6.3 D. Trigonometric Ratios of Angle in All Quadrants • Angle that are located in : • First quadrant • Second quadrant • Third quadrant • Fourth quadrant

24. Example : 1. If is an angle in the fourth quadrant and determine the values of and 2. Given and is an angle in the second quadrant, determine and 3. If and is an angle in the fourth quadrant, determine the other trigonometric values

25. y r y x x O E. Trigonometric Ratio of Related Angle a. Relate between and

26. b. Relate between and Example : Determine the values :

27. c. Relate between and Example : Determine the values :

28. d. Relate between and Example : Determine the values :

29. Y X O Negative Angle Angle means the measurement of the angle is measured in the clock wise direction.

30. Example : Solve the following problems.

31. Trigonometric Ratios for Angles Greater than Example : Calculate without using a calculator.

32. F. SIMPLE TRIGONOMETRIC EQUATIONS a. If Solution :

33. Example : Determine the x values that satisfy the following Equations in the interval

34. b. If Solution :

35. Example : Determine the values that satisfy the following Equations in the interval

36. c. If Solution : Example : Determine the values that satisfy the following Equations in the interval

37. Y r y X x O Relationships Between Polar Coordinates and Cartesian Coordinates Consider : Figure 6.4 Because and , then we obtain :

38. From these result we can conclude that : If it is known that the Polar coordinates of P are , then the Cartesian coordinates of P are Next, if the Cartesian coordinates of are known, then the polar coordinates can be determined as well. From Figure 6.4, the distance from P to O is r. Because is right-angled, then :

39. H. Polar Coordinates (Enrichment) • Cartesian coordinates of point A is defined by , means that A has abscissa x and ordinate y. • Polar coordinates of point is defined by , means P has distance r (the distance from the point to origin O) and angle (the angle between the positive X axis and the line connecting the point with origin O).

40. If the Cartesian coordinates of P are , then the polar coordinates of P are , where : and Examples : • Change the coordinates of point into polar coordinates • The polar coordinates of point P are . • Determine its Cartesian coordinates.

41. Y r y X x O H. Trigonometric Identity According to the definitions of trigonometric ratios :

42. Next, note that : Prove the other trigonometric Identities : 1. 2.

43. B A I. SINE RULE C

44. C A B I. COSINE RULE

45. I. COSINE RULE C x y A B The area of a triangle :

46. Examples : • The area of triangle ABC is • The length of sides