3.11 Trigonometric functions  (Page 2/5)

Sing of trigonometric ratio

$$\sin \theta =\frac{y}{r}<0\text{as “y” is negative}$$

$$\cos \theta =\frac{x}{r}<0\text{as “x” is negative}$$

$$\tan \theta =\frac{y}{x}>0\text{as both “x” and “y” are negative}$$

Unit circle

The angle and ratios defined in reference with circle is independent of the size of circle i.e. its radius. If radius is considered to be “1”, then we link trigonometric ratios directly to the coordinates of the tip of the terminal ray. Let x,y be the coordinates of a point “A” on a unit circle. Then,

$$\Rightarrow \sin \theta =\frac{y}{r}=\frac{y}{1}=y$$

$$\Rightarrow \cos \theta =\frac{x}{r}=\frac{x}{1}=x$$

$$\Rightarrow \tan \theta =\frac{y}{x}$$

The figure below shows what these trigonometric ratios mean with reference to circle, tangent and coordinates.

Trigonometric ratios

Trigonometric functions

There are six of trigonometric ratios. In the following sub-sections, we describe each trigonometric function with corresponding domain, range and graph. In particular, we shall come to know that some of these trigonometric functions are not defined for all values of angles. Further, we shall deliberately denote angle by variable “x” – not by “θ” as conventionally denoted. This is to emphasize that angle is a real number.

Besides, domain and range, we shall also discuss periodicity and polarity of each trigonometric function. We refer a function periodic if its values are repeated after certain interval. Graphically, periodic function has a fundamental segment, which can be used to draw plot of the function by repeating that fundamental segment again and again. Mathematically, we say that f(x+T) = f(x), where T is fundamental period.

Here, we shall make use of one important rule about periodic function. If T is the period of function f(x), then period of function $af(kx\pm b)$ is $\frac{T}{\left|k\right|}$ , where a,b and k are real numbers. Important points to note that a and b do not affect period, but coefficient of x i.e. k affect period and is given by $\frac{T}{\left|k\right|}$ .

On the other hand, polarity refers to whether the function is even or odd. If f(x) = f(-x), then function is even and its plot is symmetric about y-axis. If f(x) = -f(x), then function is odd and its plot is symmetric about origin.

Sine function

For each real number “x”, there is a sine function defined as :

$$f\left(x\right)=\sin \left(x\right)$$

The plot of sin(x) .vs. x is shown here.

Sine function

The plot, here, is continuous and period is "2π". Think period of the function in term of minimum segment which can be used to extend the plot on either side. Further as sin(-x) = -sinx, sine function is an odd function. This fact is also substantiated by the fact that plot is symmetric about origin - not y-axis.

Since function holds for all values of “x”, its domain is “R”. On the other hand, the values of sine function is bounded between “-1” and “1”, inclusive of end points. Hence, domain and range of sine function are :

$$\text{Domain}=R$$

$$\text{Range}=[-\mathrm{1,1}]$$

Let us now consider sine function which is given as :

$$f\left(x\right)=A\sin \left(x\right)$$

Multiplying sine function by a constant A does not change the periodicity of function. However, it changes the maximum and minimum values of the function. The plot extends from -A to A along y-axis as against from -1 to 1 when function is not multiplied by a constant. This, in turn, changes the range of the function :