3.11 Trigonometric functions  (Page 2/5)

$$\cos \left(x\right)=0;x=\left(2n+1\right)\frac{\pi }{2},\text{where}n\in Z$$

The function cos(x) is zero for $x=\pm \frac{\pi }{\mathrm{2,}}\pm \frac{3\pi }{\mathrm{2,}}\pm \frac{5\pi }{\mathrm{2,}}\dots$ etc. In other words, the cosine function is zero for all odd multiples of “ $\pi /2$ ”. It means that tangent function is not defined for odd multiples of “ $\pi /2$ ”. Therefore, values of “x” for which cosine is zero need to be excluded from the domain set of real number set “R”. On the other hand, the values of tangent function are extended along the real number line on either side of zero. The range of the function, therefore, is “R”. Hence, domain and range of tangent function are :

$$\text{Domain}=R-\{x:x=\left(2n+1\right)\frac{\pi }{2},n\in Z\}$$

$$\text{Range}=R$$

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

Tangent function

The period of tanx is π. Multiplication of tangent function by a constant A does not change the range as in the case of sine and cosine function. The plot is always extended on either side of x-axis so that its range is R. Multiplying argument x like tan(kx), however, changes the points where function is not defined. It is now given by :

$$x=\left(2n+1\right)\frac{\pi }{\mathrm{2k}},n\in Z\}$$

Therefore, domain is now modified as :

$$\text{Domain}=R-\{x:x=\left(2n+1\right)\frac{\pi }{\mathrm{2k}},n\in Z\}$$

Cosecant function

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

$$f\left(x\right)=\mathrm{cosec}\left(x\right)$$

Again, the function is not defined for all real number “x”. Let us recall that :

$$\Rightarrow \mathrm{cosec}x=\frac{1}{\sin \left(x\right)}$$

This is a rational polynomial form, which is defined for $\sin \left(x\right)\ne 0$ . Now, sin(x) evaluates to zero for values of “x”, which appears at a certain interval given by the condition,

$$\sin \left(x\right)=0;x=n\pi ,\text{where}n\in Z$$

This means that sin(x) is zero for $x=\mathrm{0,}\pm \pi ,\pm 2\pi ,\pm 3\pi ,\dots$ etc. In other words, the sine function is zero for all integer multiples of “ $\pi$ ”. It means that cosecant function is not defined for integral multiples of “ $\pi$ ”. Therefore, values of “x”, for which sine is zero, need to be excluded from real number set “R” for defining domain of the function.

On the other hand, values of cosecant function are fall in certain intervals. We have seen that values of sine function is between “-1” and “1”, including end points. Reciprocal of these values are either lesser than “-1” or greater than “1”. Symbolically,

$$\mathrm{cosec}x\le -1$$

$$\mathrm{cosec}x\ge 1$$

Combining two intervals, using modulus function :

$$\left|\mathrm{cosec}x\right|\ge 1$$

The combined interval of the cosecant function, therefore, is :

$$\left(-\infty ,-1\right]\cup \left(\mathrm{1,}\infty \right]$$

Hence, domain and range of cosecant function are :

$$\text{Domain}=R-\{x:x=n\pi ,n\in Z\}$$

$$\text{Range}=\left(-\infty ,-1\right]\cup \left(\mathrm{1,}\infty \right]$$

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

Cosecant function

The period of cosecx is 2π. Important to note here is that function is not defined even within a periodic segment. Since cosec(-x) = -cosecx, we conclude that cosecant function is odd function in each of periodic segment. Multiplication of cosecant function by a constant A changes the range as plot lies on or beyond -A or A. The range is modified as :

$$\text{Range}=\left(-\infty ,-\mathrm{A}\right]\cup \left(\mathrm{A,}\infty \right]$$

Multiplying argument x like cosec(kx), however, changes the points where function is not defined. It is now given by :

$$x=\frac{n\pi }{k},n\in Z$$

Therefore, domain is now modified as :

$$\text{Domain}=R-\{x:x=\frac{n\pi }{k},n\in Z\}$$

Secant function

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