5.2 RIGHT TRIANGLE TRIGONOMETRY by happo7

VIEWS: 37 PAGES: 7

									      5.2 RIGHT TRIANGLE TRIGONOMETRY
Trigonometry is study involving triangle measurement. Right triangles are used to
study functions that involve trigonometry (trigonometric functions)

Trigonometric function is a ratio of any two sides of a right angle triangle.




                                                                Hypotenuse(r)

                   Opposite(y)




                                                                          θ
                                                 Adjacent(x)


                                      x2 + y2 = r 2    (Pythagoras theorem)

The 6 ratios defined for a right triangle are:

        y                                 x                              y                r
sin θ =   ,                  cos θ =        ,                  tan θ =     ,    csc θ =     ,
        r                                 r                              x                y
        r                         x
sec θ =                 cot θ =
        x                         y

Example 1
                                                       5
Let θ be an acute angle such that sin θ =                find:
                                                      3
 a) secθ + tan θ      b) ( cot θ ) + ( csc θ )
                                  2              2




Solution
           y
                     ( )
                           2
a) sin θ =   ⇒ 32 = 5 + x 2 ⇒ x = 2
           r
               r 3                 y  5
    ⇒ sec θ = = and ⇒ tan θ = =
               x 2                 x 2
                      3+ 5
    ⇒ sec θ + tan θ =
                         2
          x     2                4
b) cot θ = =       ⇒ ( cot θ ) =
                              2

          y      5               5
                                                         2



            r   3                9
  csc θ =          ⇒ ( csc θ ) =
                              2
              =
            y    5               5

                                   4 9 13
  ⇒ ( cot θ ) + ( csc θ ) =
             2                2
                                    + =
                                   5 5 5

Trigonometric ratios of special acute angles

The equilateral and isosceles triangles shown below illustrates how trigonometric
ratios for angles 30° 45° and 60° are derived.



                                               30°30°



                                   x            h                x



                                       x                         x
                     60°               2                         2         60°

       Note:
                         h = x 2 − ( 2 )2 =
                                     x         3x
                                               2


                 x                        3x
                          1            h       3
     sin 30° =   2
                     =      , sin 60° = = 2 =
                 x        2            x  x   2

                     x
                             1                         h    3
      cos 60° =      2
                         =     ,           cos 30° =     =
                     x       2                         x   2




                                                r
                         y



                                                        45°
                                           y

                                               y         y            y        1     2
                     sin 45° = cos 45° =         =               =         =      =
                                               r       y2 + y2       y 2        2   2
                                                               3




       Using the above two triangles, the following are obtained;

       1. sin 0° = cos 90° = 0
                              1
       2. sin 30° = cos 60° =
                              2
                               1     2
       3. sin 45° = cos 45° =     =
                                2   2
                                                3
       4. sin 60° = cos 30° =
                                               2
       5. sin 90° = cos 0° = 1
       6. tan 0° = cot 90° = 0
                                                3
       7. tan 30° = cot 60° =
                                               3
       8. tan 45° = cot 45° = 1
       9. tan 60° = cot 30° = 3

     Example 2

Find the exact values of;

a)    3 sin 45° cos 30° − cot 30°

                   π             π         π
b)      2 tan          + cos         csc
                   4             3         6

                   π         π                     π
c)     2 cos           cot       + 2 3sec
                   3         4                     6


Solution
                                                        2 3      3 6 −4 3
a) 3sin 45° cos 30° − cot 30° = 3.                       .  − 3=
                                                       2 2           4

           π             π           π        1
b) 2 tan       + cos         csc       = 2.1 + .2 = 3
           4             3           6        2

               π         π                     π     1          2 3     2 +8
c)    2 cos        cot       + 2 3sec            = 2. .1 + 2 3.     =
               3         4                     6     2           3       4
                                     4


Application of Trigonometric Functions of Acute Angles

Angle of Elevation is an angle measured upwards from the horizontal as shown
in the figure below




             θ




Angle of depression is an angle measured downwards from the horizontal as
shown below


                    θ
                                            5


  Example 3

A 5 meters ladder is resting against a wall and makes an angle of 30° with the
ground. Find the height to which the ladder will reach the wall.


         Solution



                                    5               h

                            30°


         h            1
           = sin 30° = ⇒ h = 2.5m
         5            2

 Example 4

Find the height of a building if the angle of elevation from the top of the building
changes from 60° to 30° as the observer moves 30m further from the building.

  Solution



                                                h
                    30°             60°
                      30                x


h
  = tan 60° = 3 ⇒ h = x 3
x

  h                 3          30
                          x 3 30 3
       = tan 30° =    ⇒h=    +
x + 30             3       3     3

         x 3 30 3
⇒x 3=       +
          3    3
         x 3 30 3    ⎛      3 ⎞ 30 3
⇒ x 3−      =     ⇒ x⎜ 3 −
                     ⎜        ⎟=
          3    3     ⎝     3 ⎟⎠   3
                                         6


    ⎛       3 ⎞ 30 3
⇒ x⎜ 3 −
    ⎜         ⎟=
    ⎝      3 ⎟⎠   3
     2 3 30 3
⇒ x.     =
      3       3
⇒ x = 15 3

⇒ h = 15 3. 3 = 45


   Example 5

Find the exact height h of the triangle below




                                                    hm



                   30°                  60°
                         5 3 m


  Solution




                                                    hm



                   30°                  60°
                         5 3 m                  x

h
  = tan 60° = 3 ⇒ h = x 3
x
   h                  3     x 3
         = tan 30° =    ⇒h=     +5
x+5 3                3       3
                       7


        x 3
⇒x 3=       +5
         3
    ⎛      3⎞
⇒ x⎜⎜ 3−    ⎟=5
    ⎝    3 ⎟⎠
     2 3
⇒ x.     =5
      3

    5 3
⇒x=
     2
    5 3      15
⇒h=     . 3 = = 7.5m
     2        2

								
To top