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# Week 1

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### Week 1

1. 1. Analytic Trigonometry (Week 1) Irianto Universiti Teknikal Malaysia Melaka irianto@utem.edu.my September 3, 2013 Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 1 / 26
2. 2. Content Angles and Their Measure Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 2 / 26
3. 3. Content Angles and Their Measure Right Triangle Trigonometry Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 3 / 26
4. 4. Content Angles and Their Measure Right Triangle Trigonometry Computing the Values of Trigonometric Functions of Acute Angle Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 4 / 26
5. 5. Content Angles and Their Measure Right Triangle Trigonometry Computing the Values of Trigonometric Functions of Acute Angle Trigonometric Functions of General Angles Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 5 / 26
6. 6. Angle and Their Measure Deﬁnition A ray, or half-line, is that portion of a line that starts at a point V on the line and extends indeﬁnitely in one direction. The starting point V of a ray is called its vertex. Figure : Ray Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 6 / 26
7. 7. Angle and Their Measure Angle If two rays are drawn with a common vertex, they form an angle. Initial and Terminal sides We call one of the rays of an angle the initial side and the other the terminal side. The angle formed is identiﬁed by showing the direction and amount of rotation from the initial side to the terminal side. Figure : Angle, Initial Side, and Terminal Side Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 7 / 26
8. 8. Angle and Their Measure Positive and Negative Angles If the rotation is in the counterclockwise direction, the angle is positive; if the rotation is clockwise, the angle is negative. Figure : Angle Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 8 / 26
9. 9. Angle and Their Measure Standard Position An angle is said to be in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x-axis. Figure : Standard Position Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 9 / 26
10. 10. Angle and Their Measure Quadrant When an angle is in standard position, the terminal side will lie either in a quadrant, in which case we say that lies in that quadrant, or will lie on the x-axis or the y-axis, in which case we say that is a quadrantal angle. Figure : Quadrant Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 10 / 26
11. 11. Angle and Their Measure The two commonly used measures for angles are degrees and radians. Degree One complete revolution = 3600 One quarter of a complete revolution = 900 = one right angle One degree equals 60 minutes, i.e. 10 = 60 . One minute equals 60 seconds, i.e. 1 = 60 . Radian One complete revolution 2π radians = 2πc One radian is the angle subtended at the center of a circle by an arc of the circle equal in length to the radius of the circle. 1 radian ≈ 57.2950. Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 11 / 26
12. 12. Angle and Their Measure NOTE 1800 = π radians; 10 = π 1800 radian 1 radian = 1800 π Degrees 0 30 45 60 90 180 360 Radians 0 π 6 π 4 π 3 π 2 π 2π Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 12 / 26
13. 13. Angle and Their Measure Note Let α and β be positive angles. If α + β = 900, they are complementary angles. If α + β = 1800, they are supplementary angles. Figure : Complementary and Supplementary Angles Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 13 / 26
14. 14. Right Triangle Trigonometry For any acute angle θ of a right angled triangle OAB sin θ = Opposite Hypotenuse = b c cos θ = Adjacent Hypotenuse = a c tan θ = Opposite Adjacent = b a csc θ = 1 sin θ , sec θ = 1 cos θ , cot θ = 1 tan θ Figure : Right Triangle Trigonometry Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 14 / 26
15. 15. Right Triangle Trigonometry Theorem Cofunctions of complementary angles are equal. sin β = b c = cos α; cos β = a c = sin α csc β = c b = sec α; sec β = c a = csc α tan β = b a = cot α; cot β = a b = tan α Figure : Cofunctions Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 15 / 26
16. 16. Right Triangle Trigonometry Trigonometric Ratios of Allied Angles For θ <= 900 (acute) sin (900 − θ) = cos θ cos (900 − θ) = sin θ tan (900 − θ) = cot θ cot (900 − θ) = tan θ csc (900 − θ) = sec θ sec (900 − θ) = csc θ Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 16 / 26
17. 17. Right Triangle Trigonometry PPPPPPPPPθ T-ratios sin cos tan −θ − sin θ cos θ − tan θ π/2 ± θ cos θ sin θ cot θ π ± θ sin θ − cos θ ± tan θ 3π/2 ± θ − cos θ ± sin θ cot θ 2π ± θ ± sin θ cos θ ± tan θ Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 17 / 26
18. 18. Computing the Values of Trigonometric Functions of Acute Angle Commonly Used Ratios 300 , 450 , and 600 Angles sin 600 = √ 3 2 ; cos 600 = 1 2; tan 600 = √ 3 sin 300 = 1 2; cos 300 = √ 3 2 ; tan 300 = 1√ 3 sin 450 = 1√ 2 = √ 2 2 ; cos 450 = 1√ 2 ; tan 450 = 1 Figure : Commonly Used Ratios 300 , 450 , and 600 Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 18 / 26
19. 19. Trigonometric Functions of General Angles The Signs of the Trigonometric Functions The Cartesian axes divide a plane into 4 quadrants: 00 → 900 1st quadrant 900 → 1800 2nd quadrant 1800 → 2700 3rd quadrant 2700 → 3600 4th quadrant Figure : Quadrants in Cartesian Coordinates Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 19 / 26
20. 20. Trigonometric Functions of General Angles Figure : Diagram of Trigonometric Sign in Every Quadrant NOTE Quadrantal Angles: 00, 900, 1800, 2700, 3600 Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 20 / 26
21. 21. Trigonometric Functions of General Angles Coterminal Angles Two angles in standard position are said to be coterminal if they have the same terminal side. Figure : Coterminal Angle Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 21 / 26
22. 22. Trigonometric Functions of General Angles Figure : Coterminal Angle NOTE θ is coterminal with θ ± 2πk , k is any integer. The trigonometric functions of coterminal angles are equal. Example: sin θ = sin θ ± 2πk Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 22 / 26
23. 23. Trigonometric Functions of General Angles Reference Angles Let θ denote a nonacute angle that lies in a quadrant. The acute angle formed by the terminal side of θ and either the positive x-axis or the negative x-axis is called the reference angle for θ. Figure : Reference Angle Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 23 / 26
24. 24. Trigonometric Functions of General Angles Theorem If θ is an angle that lies in a quadrant and if α is its reference angle, then sin θ = ± sin α; cos θ = ± cos α; tan θ = ± tan α csc θ = ± csc α; sec θ = ± sec α; cot θ = ± cot α where the + or - sign depends on the quadrant in which θ lies. Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 24 / 26
25. 25. References For next meeting please read C. Young (2010) Algebra and Trigonometry (second edition) Wiley pp. 586–657. Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 25 / 26
26. 26. The End Irianto (UTEM) Faculty of Engineering Technology September 3, 2013 26 / 26