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Fall 2004, Triginometry 1450-02, Week 6-7 Week 6 pp.121-148 Chapter 1. Trigonometry What is the TRIGONOMETRY o Trigonometry=Angle+ Three sides + triangle + circle. Trigonometry: Measurement of Triangles (derived form Greek language) Tri means: Three (in English), drei (in German), tri (in Russian) 3 sides n=0 Points Numbers, points n=1 Line Line coordinate, Distance, absolute value N=2 2 lines Plane coordinate system Angle, slope, circle, triangle n=3 3 lines Plane and Polar coordinate Angle, Trigonometry DEFINITION: Angle Angle: is determined by rotating a ray (half line). , , , A, B, C or A, B, C Vertex of the angle: The endpoint of the ray. (Origin, central angle). O Initial side of the angle: The starting position of the ray. Terminal side of the angle: The position after rotation Standard position: Plane coordinate system: Origin=vertex, initial side=positive x axis Positive Angle: By Counter clockwise rotation. Quadrants 1,2,3,4. Negative Angle: By clockwise rotation Degree measure: equivalent to a rotation of 1 / 360 of a complete revolution about the vertex. Radian: Measure of a central angle . Coterminal Angles: Angles and have the same initial and terminal sides. 0 One Revolution: (one full rotation), 360 or Acute angle: 0 90 / 2 Right angle: Complementary angles: Two positive angles 2 0 90 0 / 2 0 0 Obtuse angle: / 2 90 180 and have sum is / 2 , , 0 Supplementary angles: Two positive angles and have sum is , , 0 0 Reference angles: Acute angle 90 by the terminal side of and the horizontal axis x . 0 Circle: Center and Radius. A full angle of ac circle from its center equals to 360 or 2 radians. Circumference: The perimeter of a circle. l 2r 2d Central angle: Angle AOC with endpoints A, C on a circle’s circumference and vertex O . Arc of a circle: Any smooth curve joining two points of the circle by a central angle. . Length of Arc: s r . / 2r / 2 Chord of a circle: The line segment joining two points on a curve. Circular segment: A portion (shaded region) of a circle whose upper boundary is arc and whose lower boundary is a chord making a central angle . Circular sector: The entire wedge-shaped area. Dis tan ce S (length of two points in real line) Time t arc length s (length of the arc on the circle) Linear speed: time t central angle (central angle of the arc on the circle) Angular speed: time t Speed: v Batmunkh.Ts Math Graduate Student 10.07.2004 Page 1 Fall 2004, Triginometry 1450-02, Week 6-7 Some common angles in Degree measure and Radian measure degree 00 30 0 45 0 60 0 90 0 120 0 Radian 0 6 4 3 2 135 0 150 0 3 4 5 6 2 3 180 0 270 0 360 0 3 2 2 Degree and Radian vs. Earth and Sun: A full degree of an angle is 2 360 . 0 10 1day Circular motion is most important motion. It is periodic. The term Earth rotation refers to the spinning of the Earth on its axis with North Pole. One rotation takes 24 hours and is called a mean solar day. If we could see down at the Earth’s North Pole from space we would see that the direction of rotation is counterclockwise. The clockwise direction is from the South Pole. The Sun (center) is a star located at the center of our solar system. The orbit of the earth around the sun is called Earth revolution. This circular motion takes 365 days (1 year) to complete one cycle around the sun. 1 year (365) days to rotate 0 One year has 365 days like 360 . 4 seasons like 4 quadrants. Each season has 3 months. 12 months. 1 month has 30 days. Batmunkh.Ts Math Graduate Student 10.07.2004 Page 2 Fall 2004, Triginometry 1450-02, Week 6-7 Semicircle 180 0 (radian and degree) Arc and angle Arrow, bow, chord The Circle is Beauty of Shape. Maximum area for a given perimeter. Minimum perimeter for a given area. Similarity property. l L l * 2 unit circle r 1 2r 2 R 2 If you want to see molecules in your eyes, increase this until an apple. Then a medium apple goes to the Earth. Batmunkh.Ts Math Graduate Student 10.07.2004 Page 3 Fall 2004, Triginometry 1450-02, Week 6-7 rad 180 0 Convert to the radian measure 1rad Convert to the degrees measure 10 180 0 180 180 0 57.30 3.14 0 180 0 180 0 sin 2 cos 2 1 Sine, cosecant y b opp r c hyp x a adj cos r c hyp sin y b tan slope cos x a sin Cosine, secant Tangent, cotangent 1 sin 1 sec cos 1 cot tan csc degree 00 30 0 45 0 60 0 90 0 120 0 135 0 150 0 Radian 0 sin 1 3 3 2 1 2 2 3 3 2 1 2 1 3 Undef 3 4 2 2 2 2 1 5 6 1 2 cos 4 2 2 2 2 2 0 6 1 2 tan sin cos 0 3 2 1 3 1 0 3 2 2 2 1 3 3 3 1 and 2 3 2 2 2 3 3 3 2 2 Sine sin cos 1 sin 180 0 270 0 360 0 3 2 1 2 0 3 2 1 1 0 1 0 Undef 0 3 0 Domain Range y f (x) Period Odd, even functions Batmunkh.Ts Cosine cos (,) (,) 1 sin 1 sin sin( 2n) period 2 sin( ) sin 1 cos 1 cos cos( 2n) period 2 cos( ) cos Odd function Even function Math Graduate Student 10.07.2004 Tangent tan sin cos n 2 (,) tan tan( n) period tan( ) tan Odd function Page 4 Fall 2004, Triginometry 1450-02, Week 6-7 Sine function Let be an angle measured counterclockwise from the x -axis (initial side) along an arc of the unit circle. Then sin y is the vertical coordinate y of the arc endpoint. As a result of this definition, the sine function is: sin sin( 2 ) Periodic function with period 2 . Odd function sin( ) sin Pythagorean Theorem: sin 2 cos 2 1 Cosine function Let be an angle measured counterclockwise from the x -axis (initial side) along an arc of the unit circle. Then cos x is the horizontal coordinate x of the arc endpoint. As a result of this definition, the sine function is: cos cos( 2 ) Periodic function with period 2 . Even function cos( ) cos Pythagorean Theorem: sin 2 cos 2 1 Tangent function The tangent function is defined by tan sin . Other notation is tan tg cos The word “tangent” also has an important related meaning as a slope, or tangent line or tangent plane. Batmunkh.Ts Math Graduate Student 10.07.2004 Page 5 Fall 2004, Triginometry 1450-02, Week 6-7 In particular, an arc is any portion (other than the entire curve) of the circumference of a circle. An arc corresponding to the central angle is denoted . Similarly, the size of the central angle subtended by this arc (i.e., the measure of the arc) is sometimes (e.g., Rhoad et al. 1984, p. 421) but not always (e.g., Jurgensen 1963) denoted . The center of an arc is the center of the circle of which the arc is a part. An arc whose endpoints lie on a diameter of a circle is called a semicircle. For a circle of radius r, the arc length l subtended by a central angle measured in radians, then the constant of proportionality is 1, i.e., Batmunkh.Ts is proportional to Math Graduate Student 10.07.2004 , and if is Page 6