Fall 2004, Triginometry 1450-02, Week 6-7


Chapter 1. Trigonometry / Week 6
pp.121-148
  • What is the TRIGONOMETRY
  • 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 / n=1 / N=2 / n=3
Points / Line / 2 lines / 3 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). , or
  • Vertex of the angle: The endpoint of the ray. (Origin, central angle).
  • 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 axis
  • Positive Angle: By Counter clockwise rotation. Quadrants 1,2,3,4.
  • 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.
  • Coterminal Angles: Angles and have the same initial and terminal sides.
  • One Revolution: (one full rotation), or
  • Acute angle:
  • Right angle:
  • Obtuse angle:
  • Complementary angles: Two positive angles and have sum is ,
  • Supplementary angles: Two positive angles and have sum is ,
  • Reference angles:Acute angle by the terminal side of and the horizontal axis .
  • Circle: Center and Radius. A full angle of ac circle from its center equals to or radians.
  • Circumference: The perimeter of a circle.
  • Central angle:Angle with endpoints on a circle’s circumference and vertex.
  • Arc of a circle: Any smooth curve joining two points of the circle by a central angle..
  • Length of Arc:.
  • 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.
  • Speed: (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)

Some common angles in Degree measure and Radian measure

degree / / / / / / / / / / /
Radian / / / / / / / / / / /

Degree and Radian vs. Earth and Sun:

A full degree of an angle is.

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.

1 year (365) days to rotate

One year has 365 days like. 4 seasons like 4 quadrants. Each season has 3 months. 12 months. 1 month has 30 days.

Semicircle (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.

  • If you want to see molecules in your eyes, increase this until an apple. Then a medium apple goes to the Earth.

Convert to the radian measure / /
Convert to the degrees measure / /
Sine, cosecant / /
Cosine, secant / /
Tangent, cotangent / /
degree / / / / / / / / / / /
Radian / / / / / / / / / / /
/ / / / / Un-
def / / / / / Un-
def /
  • and

/ Sine
/ Cosine
/ Tangent

Domain / / /
Range / / /
Period /
period /
period /
period
Odd, even functions /
Odd function /
Even function /
Odd function
Sine function

Let be an angle measured counterclockwise from the -axis (initial side) along an arc of the unit circle. Then is the vertical coordinateof the arc endpoint. As a result of this definition, the sine function is:

  • Periodic function with period.
  • Odd function
  • Pythagorean Theorem:

Cosine function

Let be an angle measured counterclockwise from the -axis (initial side) along an arc of the unit circle. Then is the horizontal coordinateof the arc endpoint. As a result of this definition, the sine function is:

  • Periodic function with period.
  • Even function
  • Pythagorean Theorem:

Tangent function

The tangent function is defined by. Other notation is

In particular, an arc is any portion (other than the entire curve) of the circumference of a circle. An arc corresponding to the central angleis 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 radiusr, the arc lengthl subtended by a central angle is proportional to , and if is measured in radians, then the constant of proportionality is 1, i.e.,

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Batmunkh.Ts Math Graduate Student 10.07.2004