Trigonometric Functions

The idea of trigonometric functions is one of the two perspectives of trigonometric relationships, the other being trigonometric ratios defined in the context of a right triangle. This resulted in an expansion of the applications trigonometry in Calculus and Physical Science wherever the ideas of rotations and vibrations occur. Let us now look at the different trigonometric functions, their properties and Graphs.

Properties of Trigonometric Functions

Trigonometric functions are periodic functions, functions whose output values repeat at regular intervals. One complete pattern is called a cycle and a period is the length of the interval consisting of one pattern. The graphs of trigonometric functions display repeated pattern over regular intervals. The graph of sine function is given below:

Periodic Function

The graph repeats at regular intervals of 2π radians. Thus the period of sine function is 2π radians. The amplitude is half the difference between the maximum and minimum values of a periodic function. For the function f(x) = sin x, the maximum value is 1 and the minimum value is -1.  Hence the amplitude of the function = $\frac{1}{2}$ ( 1 - (-1)) = $\frac{1}{2}$ x 2 = 1.

In general for the function f(x) = a . sin(bx -c) + d

Period of f(x) = $\frac{2\pi }{b}$          Period of parent function divided by the horizontal scaling factor

Amplitude = a                                     The vertical scaling factor

Phase shift = $\frac{c}{b}$

Vertical shift = d

The period for the six trigonometric functions are

Function
Period
 sin x  2$\pi$
 cos x 2$\pi$
 tan x $\pi$
 csc x 2$\pi$
 sec x 2$\pi$
 cot x $\pi$

Six Trigonometric Functions

The six trigonometric functions are

Sine            - written as sin
Cosine         - written as cos
Tangent       - written as tan   
Cosecant      - written as csc
Secant         - written as sec
Cotangent    - written as cot 


A unit circle is a circle with the center at the origin of the coordinate plane and its radius = 1 unit. All the points on the unit circle can be mapped to a variable using periodic functions. The six trigonometric functions are defined with reference to points on a Unit circle as follows. 

Unit Circle


Suppose the terminal side of angle $\theta$ in standard position (measured in the clockwise direction from the positive x axis.) cuts the Unit circle at (x, y). The six trigonometric functions of $\theta$ are,

Sin $\theta$ = y         Cos $\theta$ = x      Tan $\theta$ = y/x,     x $\neq$ 0

Csc $\theta$ = $\frac{1}{y}$,     y $\neq$ 0   

Sec $\theta$ = $\frac{1}{x}$,     x $\neq$ 0  and 

Cot $\theta$ = $\frac{x}{y}$,      y $\neq$ 0.

Trigonometric Functions Table

The trigonometric functions can be evaluated for angles 30º, 45º and 60º using special right triangles (30, 60, 90   and 45, 45, 90). And the function values can be easily found for 0º and 90º from unit circle as the coordinates representing these angles are (1, 0) and (0, 1). These values are tabled as follows. The table can be memorized and used for evaluating trigonometric functions of angles whose reference angles are these special angles. The function values of quadrant angles 180º and 270º are also shown in the table.

Function
   0º or
0 radians
    30º or
π/6 radians
    45º or
π/4 radians
    60º or
π/3 radians
    90º or
π/2 radians
 180º or
π radians
   270º or
3π/2 radians
 Sin      0
     1/2
    √2/2     √3/2       1
     0
      -1
 Cos      1
    √3/2     √2/2       1/2
      0
    -1
       0
 Tan      0
    √3/3        1
      √3  Undefined     0
 Undefined


The reciprocal functions Csc, Sec and Cot can be found by taking the reciprocals of values given in the table.

Trigonometric Functions of Any Angle

We saw how the six trigonometric functions can be found using Unit circle. Using only the coordinate plane, the six trigonometric functions can be found using the following

Definitions:

Let (x, y) be a point on the terminal side of the angle $\theta$ in standard position and r = $\sqrt{x^{2}+y^{2}}$ $\neq$ 0, then

Sin $\theta$ = $\frac{y}{r}$

Cos $\theta$ = $\frac{x}{r}$

Tan $\theta$ = $\frac{y}{x}$,  x $\neq$ 0
Any Angles
Csc $\theta$ = $\frac{1}{y}$,  y $\neq$ 0

Sec $\theta$ = $\frac{1}{x}$,  x $\neq$ 0

Cot $\theta$ = $\frac{x}{y}$,   y $\neq$ 0

Let us solve one example using the above definition:

Let (-4, 3) be a point on the terminal side of angle $\theta$.  Evaluate Sin $\theta$, Cos $\theta$ and Tan $\theta$.

x = -4 and y = 3. The terminal side of angle $\theta$ falls in the second quadrant.

r = $\sqrt{x^{2}+y^{2}}$ = $\sqrt{(-4)^{2}+(3)^{2}}=\sqrt{16+9}$ = $\sqrt{25}$ = 5


Hence Sin $\theta$ = $\frac{y}{r}$ = $\frac{3}{5}$

          Cos $\theta$ = $\frac{x}{r}$ = $\frac{-4}{5}$    and

          Tan $\theta$ = $\frac{y}{x}$ = $\frac{3}{-4}$ = $\frac{-3}{4}$.

The trigonometric functions of any angle can also be evaluated using reference angle. Reference angle is an acute angles made by the terminal side of any angle θ with the horizontal axis. The formulas for finding reference angle θ' of an angle θ are given below

θ' = θ                    for 0 ≤ θ ≤ 90º or  0 ≤ θ ≤ π/2 radians         First quadrant angles

θ' = 180 - θ           for 90º ≤ θ ≤ 180º  or
θ' = π - θ                    π/2 ≤ θ ≤ π radians          Second quadrant angles  

θ' = 180 + θ          for 180º ≤ θ ≤ 270º    or
θ' = π + θ                     π ≤ θ ≤ 3π/2 radians      Third quadrant angles

θ' = 360 - θ          for 270º ≤ θ ≤ 360º    or
θ' = 2π - θ            for 3π/2 ≤ θ ≤ 2π radians       fourth quadrant angles.

Method to evaluate trigonometric functions using reference angle
  1. Find the reference angle θ' using the appropriate formula given above.
  2. Find the function value for θ' either using the trigonometric tables or calculator
  3. Determine the sign of the function using quadrant rules. Affix the sign to the function value found in step 2.

To evaluate trigonometric functions for angles greater than 360º, the multiples of 360 are to be subtracted from the angle first and then evaluate either using Unit circle or reference angle.

Trigonometric Functions Problems

Below you could see trigonometric functions problems

Solved Examples

Question 1: Evaluate Sin (-3300)
Solution:
 
Sin (-3300) = - Sin 3300     Sin (-$\theta$) = -Sin $\theta$

    The terminal side of angle 330º falls in the quadrant IV and the reference angle of 3300 = 360 - 330 = 300.

    In the fourth quadrant Sine function values are all negative.

    Sin 3300 = -Sin 300 = -$\frac{1}{2}$      Using the Table of Trigonometric values

    Hence Sin (-3300) = - Sin 3300 = -(-$\frac{1}{2}$) = $\frac{1}{2}$.

 

Question 2: Given Sin $\theta$ = $\frac{3}{5}$  and $\frac{\pi}{2}$ ≤ $\theta$ ≤ 2$\pi$. Evaluate the other trigonometric functions of $\theta$.
Solution:
 
We can find Cos $\theta$ using Pythagorean identity Sin2 $\theta$ + Cos2 $\theta$ = 1  

    Cos2 $\theta$ = 1 - Sin2 $\theta$  = 1 - $(\frac{4}{5})^{2}$ =  $\frac{16}{25}$

    Cos $\theta$ = -$\frac{4}{5}$                      As Cos $\theta$ is negative in Quadrant II.

    Tan $\theta$ = $\frac{Sin\ θ}{Cos\ θ}$           Quotient Identity

             = -$\frac{3}{4}$

    The other reciprocal identities are

    Csc $\theta$ = $\frac{1}{Sin\ θ}$ = $\frac{5}{3}$

    Sec $\theta$ = $\frac{1}{Cos\ θ}$ = -$\frac{5}{4}$

    Cot $\theta$ = $\frac{1}{Tan\ θ}$ = -$\frac{4}{3}$.

 

Trigonometric Functions Graphs

We saw the graph of sin x when we discussed about the properties of Trigonometric functions.  In a similar manner we can find the patterns repeated in the graphs of other trigonometric functions for every period interval.

The Graph of Cos function is given below:

Cosine Graph

The Period of cosine graph is also 2π as for as sine graph. Both functions have continuous wave like graph. They differ only in x and y intercepts. Sine graph can also be viewed as a,

Cos graph with a phase shift = $\frac{\pi}{2}$ as both are co-functions of each other.   Sin x = Cos ($\frac{\pi}{2}$ - x) = Cos (x - $\frac{\pi}{2}$).

Sine and Cosine functions model many real life phenomenon.


Let us all also view the graph of tan function.

Tan Graph

We see the repeated patterns in the graph for every interval length of $\pi$ units. Thus the period of tan function = $\pi$. Unlike the sine and cosine graphs, the patterns repeat with breaks at every odd multiple of $\frac{\pi}{2}$ as the tan function is undefined for these values. The pattern repeats with infinite jumps at every odd multiple of $\frac{\pi}{2}$.

Topics in Trigonometric Functions