Chapter 1. Trigonometry Week 6 pp

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1 Fall, Triginometry 5-, Week -7 Chapter. Trigonometry Week pp.-8 What is the TRIGONOMETRY o TrigonometryAngle+ Three sides + triangle + circle. Trigonometry: Measurement of Triangles (derived form Greek language) Tri means: Three (in English), drei (in German), tri (in Russian) sides n n N n Points Line lines lines Numbers, points Line coordinate, Distance, absolute value Plane coordinate system Angle, slope, circle, triangle Plane and Polar coordinate Angle, Trigonometry DEFINITION: Angle Angle: is determined by rotating a ray (half line). α β, γ,, 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: Originvertex, initial sidepositive x axis Positive Angle: By Counter clockwise rotation. Quadrants,,,. Negative Angle: By clockwise rotation Degree measure: equivalent to a rotation of / of a complete revolution about the vertex. Radian: Measure of a central angle θ. Coterminal Angles: Angles α and β have the same initial and terminal sides. or One Revolution: (one full rotation), Acute angle: < θ < 9 / Right angle: θ 9 / Obtuse angle: / 9 < θ < 8 Complementary angles: Two positive angles α and β have sum is α + β /, α, β Supplementary angles: Two positive angles α and β have sum is α + β, α, β < θ 9 by the terminal side of θ and the horizontal axis x. Reference angles: Acute angle Circle: Center and Radius. A full angle of ac circle from its center equals to or radians. Circumference: The perimeter of a circle. l r d Central angle: Angle AOC with endpoints A, C on a circle s circumference and vertexo. Arc of a circle: Any smooth curve joining two points of the circle by a central angle.. Length of Arc: s r θ. θ / r θ / 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 θ <. Circular sector: The entire wedge-shaped area. Speed: Dis tan ce S v Time t arclength s time t central angle θ time t (length of two points in real line) Linear speed: (length of the arc on the circle) Angular speed: (central angle of the arc on the circle) Batmunkh.Ts Math Graduate Student.7. Page

Fall, Triginometry 5-, Week -7 Some common angles in Degree measure

Earth and Sun: A full degree of an angle is.

The term Earth rotation refers to the spinning of the Earth on its

One rotation takes hours and is called a mean solar day.

see that the direction of rotation is counterclockwise.

year (5) days to rotate One year has 5 days like month has days.

2 Fall, Triginometry 5-, Week -7 Some common angles in Degree measure and Radian measure degree Radian 5 Degree and Radian vs. Earth and Sun: A full degree of an angle is. day 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 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. year (5) days to rotate One year has 5 days like month has days. 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 5 days ( year) to complete one cycle around the sun.. seasons like quadrants. Each season has months. months. Batmunkh.Ts Math Graduate Student.7. Page

3 Fall, Triginometry 5-, Week -7 Semicircle 8 (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. l r L R Similarity property. unit circle r If you want to see molecules in your eyes, increase this until an apple. Then a medium apple goes to the Earth. l * Batmunkh.Ts Math Graduate Student.7. Page

4 Fall, Triginometry 5-, Week -7 rad 8 Convert to the radian measure rad Convert to the degrees measure θ θ 8 θ θ 8 sin θ + cos θ Sine, cosecant y b opp r c hyp x a adj r c hyp y b tan θ slope x a Cosine, secant Tangent, cotangent sec θ cot θ tan θ csc θ degree Radian 5 tan θ cosθ Undef and Sine Undef Domain Range θ y f (x ) Period Odd, even functions Batmunkh.Ts Cosine Tangent tan θ (,+ ) (,+ ) sin( θ + n) period sin( θ ) cos(θ + n) period cos( θ ) ± n (,+ ) tan θ tan( θ + n) period tan( θ ) tan θ Odd function Even function Odd function Math Graduate Student.7. θ Page

5 Fall, Triginometry 5-, Week -7 Sine function Let θ be an angle measured counterclockwise from the x -axis (initial side) along an arc of the unit circle. Then y is the vertical coordinate y of the arc endpoint. As a result of this definition, the sine function is : Periodic function with period. sin( θ + ) Odd function sin( θ ) Pythagorean Theorem: sin θ + cos θ Cosine function Let θ be an angle measured counterclockwise from the x -axis (initial side) along an arc of the unit circle. Then x is the horizontal coordinate x of the arc endpoint. As a result of this definition, the sine function is: Periodic function with period. cos(θ + ) Even function cos( θ ) Pythagorean Theorem: sin θ + cos θ Tangent function The tangent function is defined by tan θ. Other notation is tan θ tgθ The word tangent also has an important related meaning as a slope, or tangent line or tangent plane. Batmunkh.Ts Math Graduate Student.7. Page 5

6 Fall, Triginometry 5-, Week -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. 98, p. ) but not always (e.g., Jurgensen 9) 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 is proportional to, and if is measured in radians, then the constant of proportionality is, i.e., Batmunkh.Ts Math Graduate Student.7. Page

Math 1205 Trigonometry Review

Math 1205 Trigonometry Review Math 105 Trigonometry Review We begin with the unit circle. The definition of a unit circle is: x + y =1 where the center is (0, 0) and the radius is 1. An angle of 1 radian is an angle at the center of