Chapter 1 : Introduction to Trigonometry

Topics covered in this snack-sized chapter:



Introduction to Trigonometry arrow_upward


  • Trigonometry is a branch of mathematics that deals with triangles, mostly right triangles.
  • It defines relationships among sides and angles of triangles.


    • Triangle Properties: Similar Triangles arrow_upward


  • Same Shape.
  • Corresponding angles are congruent.
  • Corresponding sides are in proportion.
  • Example:

  • For  and ,
  • Similarity ratio is given by:
  • Therefore .

    • Solving the Height Problem (Using Similar Triangles) arrow_upward


  • Example: Johnson is  tall, but his shadow is only. long. There is a tree across the street with a shadow of.
  • The sun hits the tree and Johnson, at the same angle to make the shadows. How tall is the tree?

  • Solution:
  • Let the height of tree be
  • Using Similar Triangles (proportional sides).
  •  For the  and :

    •      

  • That is:

  • We get,

    •    

       25 x = 600

       x = 24

  • Hence, height of the tree is

    • Angles and Rotation arrow_upward


  • An angle can be thought of as a “rotating ray.”
  • The angle’s measure is generated by a rotation about the angle’s vertex, from the initial side to the terminal side.
  • An angle is in standard position if
    • The vertex is at the origin of the and  axis.
    • The initial side of the angle lies along the positive axis.

  • Quadrant Angles lie on the  or  axis
    • Counter Clockwise rotation  Positive angle.
    • Clockwise rotationNegative angle.

    Ray arrow_upward


  • Aray, or half-line, is that portion of a line that starts at a point  on the line and extends indefinitely in one direction.
  • The starting point  of a ray is called its vertex.

  • If two lines are drawn with a common vertex, they form an angle.
  • One of the rays of an angle is called the initial side and the other is terminal side.


    • Measuring Angles arrow_upward


  • Angles are commonly measured in:
    • Degrees
    • Radians
  • One complete rotation of a ray is.
  • Minutes and Seconds measure portions of a degree.


    • 360 Degree Angle arrow_upward


  • The angle formed by rotating the initial side exactly once in a counterclockwise direction until it coincides with itself is said to measure  abbreviated.

  • One degree is  of a revolution.

    • 90 Degree Angle arrow_upward


  • A right angle is an angle of or   revolution.

    •  

  • angle is of a revolution.

    • 180 Degree Angle arrow_upward


  • A straight angle is an angle of , or  revolution.

  •  angle is of a revolution.

    • One Radian arrow_upward


  • Consider a circle of radius.
  • Construct an angle:
    • Whose vertex is at the center of this circle, called the central angle.
    • Whose rays subtend an arc on the circle whose length is .
  • The measure of such an angle is 1 radian.


    • Arc Length Theorem arrow_upward


  • For a circle of radius, a central angle of  radians subtends an arc whose length  is .
  • Suppose an object moves along a circle of radius  at a constant speed.
  • If  is the distance traveled in time  along this circle, then the linear speed  of the object is defined as:

    •    

  • As the object travels along the circle, suppose that   is the central angle swept out in time.
  • Then the angular speed  of this object is the angle swept out divided by the elapsed time.

    •    

       


    Relationship between Radians and Degrees arrow_upward


     

     

     


    Angles in the Coordinate Plane arrow_upward


  • 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 positive axis.
    • Positive angle rotate counter clockwise:

  • Negative angle rotates clockwise:


    • Types of Angles arrow_upward


  • An acute angle measures less than.

  • A right angle, one quarter of a complete rotation, measures.

  • An obtuse angle measures more than  but less than .

  • A straight angle measures.



    • Thank You from Kimavi arrow_upward


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