In the previous section, we saw the domain and range of cosine and secant functions. In this section, we will see the domain and range of tangent and cotangent functions.
Let us find the domain and range of f(x) = tan x. It can be written in 2 steps:
1. First we will find the domain:
• We have seen that:
♦ In the case of f(x) = tan x,
♦ $\frac{(2n+1)\pi}{2}$ where n is any integer, should not be used as input x
• So we can write:
Domain of f(x) = csc x is $R-\{x:x=\frac{(2n+1)\pi}{2},\; n\, \in \,Z\}$
• That means, we must subtract the set $\{x:x=\frac{(2n+1)\pi}{2},\;
n\, \in \,Z\}$ from R. The resulting set after subtraction, is the
domain of f(x) = tan x
• $\{x:x=\frac{(2n+1)\pi}{2},\; n\, \in \,Z\}$ is the set containing all $\frac{(2n+1)\pi}{2}$, where n is any integer.
2. Let us find the range of f(x) = tan x
• The range in this case can be better understood if we analyze the graph of f(x) = cos x and f(x) = sec x and f(x) = tan x together. It is shown in fig.3.32 below:
 |
| Fig.3.32 |
(i) Consider the segment from 0 to $\frac{\pi}{2}$ on the x axis.
• In this segment,
♦ the sine curve rises
♦ the cosine curve falls
♦ the tangent curve rises
• Let us see the reason for the rise in tangent curve:
♦ We have: $\tan x= \frac{\sin x}{\cos x}$
♦ The rise in sine curve indicates increase in numerator.
♦ The fall in cosine curve indicates decrease in denominator.
♦ So the fraction $\frac{\sin x}{\cos x}$ will obviously increase.
• At zero angle, sine is 0 and cosine is 1
♦ So tangent will be zero. This is clearly visible in the graph
• At $\frac{\pi}{2}$ angle, sine is 1 and cosine is 0
♦ So tangent can not be defined.
♦ Indeed we see that, the tangent curve do not touch the vertical line through $\frac{\pi}{2}$
• As the angle approaches $\frac{\pi}{2}$, the cosine value becomes very small and consequently the tangent becomes very large.
| ◼ As seen before, we cannot put x = $\frac{\pi}{2}$ in $f(x) = \tan x = \frac{\sin x}{\cos x}$. We can write: • As x approaches $\frac{\pi}{2}$, cos x becomes smaller and smaller, getting closer and closer to zero. (For example, values like 0.00001, 0.000001 are very close to zero) • As cos x becomes smaller and smaller, tan x becomes larger and larger. • This is indicated by the rising portion of the tangent curve between 0 and $\frac{\pi}{2}$. • As angle becomes closer and closer to $\frac{\pi}{2}$, this rising portion gets closer and closer to the vertical line through $\frac{\pi}{2}$. (why does this happen? The reader may write the answer in his/her own notebooks) • But it never touches that vertical line. • If it touch, it would mean that, x = $\frac{\pi}{2}$ is a point in the tangent curve. We know that, it cannot happen. |
(ii) In a similar way, the reader must be able to analyze the remaining segments:
♦ $\frac{\pi}{2}$ to π
♦ π to $\frac{3\pi}{2}$
♦ $\frac{3\pi}{2}$ to 2π
(iii) When all the segments are analyzed, we get a good understanding about the pattern.
3. We see that the tangent curves extend from very high up to very low bottom in the graph. There is no break. That means, all values in the y axis are present in the curves.
• We can write: Range of f(x) = tan x is R
Let us find the domain and range of f(x) = cot x. It can be written in 2 steps:
1. First we will find the domain:
• We have seen that:
♦ In the case of f(x) = cot x,
♦ nπ, where n is any integer, should not be used as input x
• So we can write:
Domain of f(x) = cot x is $R-\{x:x=n \pi,\; n\, \in \,Z\}$
• That means, we must subtract the set $\{x:x=n \pi,\; n\, \in \,Z\}$ from R. The resulting set after subtraction, is the
domain of f(x) = cot x
• $\{x:x=n \pi,\; n\, \in \,Z\}$ is the set containing all nπ, where n is any integer.
2. Let us find the range of f(x) = cot x
• The range in this case can be better understood if we analyze the graph of f(x) = tan x and f(x) = cot x together. It is shown in fig.3.33 below:
 |
| Fig.3.33 |
(i) Consider the segment from 0 to $\frac{\pi}{2}$ on the x axis.
• In this segment,
♦ the tangent curve rises
♦ the cotangent curve falls
♦ this is obvious because, cot is the reciprocal of tan
• At zero angle, tan is zero
♦ The reciprocal of zero does not exist.
♦ So at zero angle, the reciprocal of tan is not defined.
♦ Indeed we see that, the cot curve do not touch the vertical line through 0
(ii) The reader must be able to explain the whole pattern of cot in this way.
(iii) Note that, the sign does not change in the reciprocal
♦ If tan is +ve in a segment, cot will also be +ve in that segment.
♦ If tan is -ve in a segment, cot will also be -ve in that segment.
3. We see that the cotangent curves extend from very high up to very low bottom in the graph. There is no break. That means, all values in the y axis are present in the curves.
• We can write: Range of f(x) = cot x is R.
In
the next
section, we will see trigonometric identities.
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