Trigonometric Fuctions

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TRIGONOMETRIC FUNCTIONS
Trigonometric functions can be defined in two different but equivalent ways :
(a) Trigonometric Functions of Real Numbers &
(b) Trigonometric Functions of Angles.
First we shall discuss trigonometric functions of real numbers.

                                THE UNIT CIRCLE
The unit circle is the circle of radius 1 and centre at the origin in the x-y plane. Its
equation is
                                       x 2 + y2 = 1




                                 ww
                                  w
                                  ww
                                   w g
                            f    pff 1f
                                 f3 f f
                                  ff
                                   ff
                                  fff f1f 3f
                                              f     g
                                 and Q f, f on the unit circle?
                                        f f
Example : Are the points P @   ,
                             2 2       2 2

Solution :
Any point on the unit circle will satisfy the equation x 2 + y 2 = 1 A
Since
        w
        w
        w
        ww
       w g2   g2
 f     pff f 1f
       f3 f
        ff
       fff
        ff    f
              f        3f 1f
   @         +           f f
                        f f
                    = f+ f= 1
      2        2       4 4
P is on the unit circle A

       f g2 f g2
        1f
         f
         f   3f
              f
              f            1f 9f 10f
Since        +              f f f
                            f f fff
                         = f+ f= ff≠ 1
        2        2         4 4 4
Q is not on the unit circle A

               1f
                 f         g
                f
Example : If P f, y is on the unit circle, what is its y @ coordinate?
               3

Solution :
                 f g2
                  1f
                   f
                   f
                          + y 2 = 1,
                     3
                          1f 8f
                           f f
                           f f
                 y2 = 1 @ = ,
                          9 9
                        w
                        w
                        w
                        ww
                         w
                         w
                         w        w
                                  ww
                                   ww
                                    w
                                 p2
                 y =F  s 8f= F 2ffff
                          ff     ffff
                                 ffff
                                fffff
                         9        3

Terminal Points on the Unit Circle :

Let t be a real number. We start at the point (1,0) and mark of a distance t along the unit
circle. If t is positive, we proceed in the counterclockwise direction and if t is negative,
we proceed in the clockwise direction. Suppose we reach a point P(x,y) on the unit circle.
This point P(x,y) reached this way is called the terminal point associated with the real
number t .

The circumference of the unit circle is C = 2π 1 = 2π . So, for one full rotation the
                                                      ` a

distance covered will be 2π .

Moving counterclockwise : If a point starts at (1,0) and move counterclockwise and
                                          .
come back to (1,0), it moves a distance 2πb c
      t = 0 determines the terminal point 1,0 A
       t = 2π also determines the terminal point 1,0 A
                                                  b       c


       t = π determines the terminal point @ 1,0 A
                                          b       c

           πf
            ff
       t = ffdetermines the terminal point 0,1
                                           b c
           2
           3πf
            ff
            ff
             f
       t = ff determines the terminal point 0, @ 1 A
                                            b      c
            2

Moving clockwise : t is negative here. If a point starts at (1,0) and move clockwise and
come back to (1,0), it moves a distance 2πb c
                                          .
      t = 0 determines the terminal point 1,0 A
       t = @ 2π also determines the terminal point 1,0 A
                                                      b       c


       t = @ π determines the terminal point @ 1,0 A
                                              b       c

             πf
              ff
       t = @ ffdetermines the terminal point 0, @ 1
                                             b     c
             2
             3πf
              ff
              ff
               f
       t = @ ff determines the terminal point 0,1 A
                                              b c
              2
                 TRIGONOMETRIC FUNCTIONS OF
                       REAL NUMBERS

There are six trigonometric functions, which are sine, cosine, tangent, cotangent, secant
and cosecant and are denoted by symbols sin, cos, tan, cot, sec and csc respectively. The
trigonometric functions of the real number t are defined below :

Let t be a real number and let P(x,y) be the terminal point on the unit circle associated
with t . Then the trigonometric functions of the real number t are :
                                                                 yf
                                                                  f
                                                                 fff
        sin t = y,                cos t = x,              tan t = ,x ≠ 0
                                                                 x
                 xf
                 fff
                   f                      1f
                                           f
                                           ff                     1f
                                                                   f
                                                                  fff
        cot t = , y ≠ 0           sec t = , x ≠ 0         csc t = , y ≠ 0
                 y                        x                       y
        These are also called trigonometric ratios A

                               3f 4f
                          f        g
                                 f ff
Example : If the point P @ f, f on the unit circle corresponds to the real number t,
                               5 5
find the six trigonometric functions of t .

Solution :
           3f      4f
             f
Here, x = @ f, y = fff
           5       5
                                                                             4f
                                                                             ff
                                                                              ff
           4f
            f
            ff                              3f
                                             ff                        yf fff
                                                                       ff fff
                                                                        f fff
                                                                        f ff    4f
                                                                                 f
                                                                                 ff
sin t = y = ,                  cos t = x = @ ,                  tan t = = 53 = @ ,
           5                                5                          x @  f
                                                                            f
                                                                            ff  3
                                                                              5
             3f
             ff
              f
       xf @ff
         f f5 f
        f fff
       ff ffff  3f
                 f
                 ff                        1ff 5f
                                          fff
                                          fff
                                           ff
                                           f      f f                    1f 5f
                                                                          f f
                                                                          f f
                                                                         ff f
cot t = = 4 = @                 sec t =        =@                csc t = 4 =
       y   f
           ff
            ff  4                         @
                                            3f
                                            fff   3                      f 4
                                                                          f
                                                                          f
                                                                          ff
           5                                5                           5

                                       ANGLES
                  jj
                  jj
                  j
                  jj
                   j
                   j
                   j
                   k
                   j
Let us take a ray OA with its vertex (endpoint) as O. If we rotate it about the vertex O ,
                               jj
                               jj
                               jj
                                j
                                j
                                jk
                                 j
                                 j
and come to the final position OB , the trigonometric angle is the measure of rotation or
                                                        jj
                                                         jj
                                                         jj
                                                         j
                                                         jj
                                                          k
                                                          j
                                                          j
amount of rotation. The starting position of the ray or OA is called the initial side and
                       jj
                       jj
                       j
                       jj
                        j
                        j
                        j
                        k
                        j
the ending position or OB is called the terminal side.
If the rotation of the ray from its initial position is in the counterclockwise direction, the
angle is assigned a positive sign. If it has rotated in the clockwise direction, the angle is
assigned a negative sign.

A zero angle corresponds to zero rotation, in which case the initial and the terminal sides
are coincident. The two sides also be come coincident for any number of complete
rotations in the positive or negative direction.

                             MEASURE OF ANGLES
In Trigonometry and in Calculus, the angles are normally measured in radian measure.

Radian Measure of angles : If in a circle we take an arc of length equal to the radius of
that circle then the angle it subtends at the centre of the circle is called 1 radian.

Since the circumference of radius r is of length 2π r , a full revolution in the
counterclockwise direction has a measure of 2π radians.

Degree measure of angles : Angles are also measured in degrees ° . A positive angle of
                                                                       ` a

one full rotation has a measure of 360 ° . Thus,
                               2π radians = 360 °
                                π radians = 180 °
To transform degree measures in radian measures and vice versa we can use
                                     πff
                                     fff
                                      ff
                                      f
                                     fff
                               1°=      radian
                                    180
                                          180f
                                           fff
                                            ff
                                            ff
                                           fff
                               1 radian =      °
                                           π

Some common angles in both the measures :

Degrees      0°       30 °      45 °     60 °      90 °     120 °     180 °    270 °     360 °
Radians      0         πf
                       ff
                        ff       πf
                                  f
                                 fff      πf
                                           f
                                          fff       πf
                                                     f
                                                    fff      2πf
                                                               f
                                                              ff
                                                              ff
                                                               ff       π       3πf
                                                                                  ff
                                                                                  f
                                                                                 ff
                                                                                 ff       2π
                       6         4        3         2         3                  2
Quadrants                         I                           II             III      IV

                                                 2πf
                                                  ff
                                                  ff
                                                   f
Eaxmple : Transform (a) 225 ° in radians and (b) ffradians in degrees.
                                                  3

Solution :
                  πff ff
                 fff 5πf
                  ff ff
                  f    f
 a 225 ° = 225 B fff= ffradians
` a
                180 4
` a 2πf         2πf 180f
     ff
     ff
      f
 b ffradians = ffB fff = 120 °
                 ff fff
                 ff ff
                  f   ff
                        °
     3           3   π


                                                πf
      •   An angle of measure between 0 to ff    ff
                                                   radians (between 0 ° and 90 ° ) is called an
                                                2
          acute angle.
                                πf
      •   An angle of measure ff  f
                                  f
                                    radians ( 90 ° ) is called a right angle.
                                2
                                              πf
      •                                        ff
          An angle of measure between ffto π radians (between 90 ° and 180 ° ) is
                                              2
          called an obtuse angle.
      •   An angle of measure π radians ( 180 ° ) is called a straight angle.
                                                                πf
      •   If α and β are two angles that α + β = ff ,             ff
                                                                         α and β are called
                                                                 2
          complementary angles.
      •   If α and β are two angles that α + β = π ,                     α and β are called
          supplementary angles.
                                                                  πf
                                                                   f
Example : Find an angle complementary to (a) a θ = 45.4 ° , b θ = fff
                                             ` a           ` a
                                                                  6

Solution :
  a The complementary angle of θ is 90 ° @ θ = 90 ° @ 45.4 ° = 44.6 ° A
 ` a

                                    πf       πf πf πf
                                     f
                                     f        f f f
                                              f f f
  b The complementary angle of θ is ff θ = ff ff ff
                                               @ =
 ` a
                                      @
                                    2        2 6 3

                      TRIGONOMETRIC FUNCTIONS
                             OF ANGLES

An angle is in standard position in the Cartesian coordinate system, when its initial side
corresponds to the positive x-axis and the vertex is at the origin. In the picture below
angleθ is in standard position and the point P(x,y) denotes a point on the terminal side of
θ.
In the figure above, the Quadrants are represented by the numbers I, II, III, IV.
                                                                wwww
                                                                 wwww
                                                                 wwww
                                                                 wwww
                                                                 wwww
                                                                  wwww
                                                                  wwww
                                                                wwww
The distance r between the origin and the point P (x,y) is   r= qx 2 + y 2


There are six trigonometric functions, which are sine, cosine, tangent, cotangent, secant
and cosecant and are denoted by sin, cos, tan, cot, sec and csc respectively. These
functions for the above angle t are defined as follows :

Let the point P(x,y) denotes a point on the terminal side of angle θ and the distance of P
from the origin be r , then the trigonometric functions or ratios of the angle θ are :
        yf
        ff
         ff                         xf
                                     ff
                                      f
sin θ = ,                   cos θ = ,
        r                           r
        yf
         ff
         ff                         xf
                                     f
                                    fff
tan θ = ,x ≠ 0              cot θ = , y ≠ 0
        x                           y
         rf
         ff
          f                          rf
                                      f
                                     fff
sec θ = , x ≠ 0              csc θ = , y ≠ 0
        x                            y

                                                 1           1 ff        1ff
                                                ffff
                                                fff
                                                 fff        fff
                                                            fff
                                                             f
From the definition, it is obvious that secθ = ffff cscθ = fff, cotθ = ffff and also
                                                    ,                    f
                                                                        fff
                                                                        fff
                                                                            ,
                                               cosθ        sinθ        tanθ
        sinθf          cosθf
         ffff
         fff
          ff
 tanθ = ffff cotθ = ffff
             ,           fff
                         fff
                        fff
                             A
        cosθ           sinθ

The above trigonometric ratios remain same for any point P taken on the terminal side of
angle θ. There are four quadrants as shown in the above figure. These are same as the
quadrants described in coordinate geometry. The angle is in a particular quadrant
(quadrant II in the above diagram), when the terminal side of the angle falls in that
quadrant.

To remember the signs of different trigonometric functions we can remember the
following picture :
    •   In Quadrant I : All the six trigonometric functions i.e. sinθ to cscθ are positive,
        as x and y are both positive there.
    •   In Quadrant II : Only two functions sinθ & its reciprocal cscθ are positive, and
        rest all are negative there ( as x is negative and y is positive there).
    •   In Quadrant III : Only two functions tanθ & its reciprocal cotθ are positive, and
        rest all are negative there (as x and y are both negative there).
    •   In Quadrant IV : Only two functions cosθ & its reciprocal secθ are positive, and
        rest all are negative there ( as y is negative and x is positive there).

In calculators, we get approximate values of trigonometric ratios in decimals. However,
by using geometric properties, we can find the exact functional values of some special
angles, as given in the table below :

b
   α c    α      sinα  cosα     tanα     cotα      secα      cscα
 degree radian
        `      a

 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
   0°     0        0     1        0    undefined      1    undefined
                       ww
                        w
                        ww
                         w      ww
                                 w
                                 w
                                 ww                   ww
                                                       ww
                                                        w
                                                        w
           πf
           ff
            ff     1f
                    f
                    f  pff
                       f3 f
                         ff
                        ff
                        fff    pff
                                f3 f
                                 ff
                                 ff
                                fff       w
                                          w
                                          ww
                                           w
                                           w         pff
                                                     ff3 f
                                                   2ffff
                                                     fff
                                                    fffff
  30 °                                   p3                     2
           6      w
                  w
                  w
                  w
                   2
                   w
                   w
                        2w
                        w
                        ww
                         ww
                                 3                    3
           πf
           ff
            ff   pff
                 f2 f
                  fff
                  ff
                   ff  pff
                        f2 f
                         ff
                        fff
                         ff                          w
                                                     ww
                                                      w
                                                      ww       w
                                                               ww
                                                                w
                                                                w
                                                                w
  45 °                           1         1        p2        p2
           4      2w
   
				
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