Discuss the continuity of the cosine, cosecant, secant, and cotangent functions
Solution:
It is known that if g and h are two continuous functions, then
(i) h(x) g(x), g(x) ≠ 0 is continuous.
(ii) 1g(x),g(x) ≠ 0 is continuous.
(iii) 1h(x), h(x) ≠ 0 is continuous.
Let g(x) = and h(x) = cos x are continuous functions.
It is evident that g(x) = sin x is defined for every real number.
Let c be a real number.
Put x = c + h
If x→c, then h→0
g(c) = sin c
limx→c g(x) = limx→c sinx = limh→0 sin(c + h)
= limh→0 [sin c cos h + cos c sin h]
= limh→0 (sin c cos h) + limh→0 (cos c sin h)
= sin c cos 0 + cos c sin 0
= sin c (1) + cos c (0)
= sin c
⇒ limx→c g(x) = g(c)
Therefore, g(x) = sinx is a continuous function.
Let h(x) = cosx
It is evident that h(x) = cos x is defined for every real number.
Let c be a real number. Put x = c + h
If x→c, then h→0
h(c) = cos c
limx→c h(x) = limx→c cos x
= limh→0 cos(c + h) = limh→0 [cos c cos h − sinc sin h]
= limh→0 (cosc cosh) − limh→0 (sin c sin h)
= cos c cos 0 − sinc sin 0
= cos c(1) − sin c(0)
= cos c
⇒ limx→c h(x) = h(c)
Therefore, h(x) = cos x is a continuous function.
Therefore, it can be concluded that,
cosec x = 1/sin x, sin x ≠ 0 is continuous.
⇒ cosec x, x ≠ nπ (n ∈ Z) is continuous.
Therefore,
cosecant is continuous except at x = nπ (n ∈ Z)
sec x = 1/cos x, cos x ≠ 0 is continuous.
⇒ sec x, x ≠ (2n + 1) π / 2 (n ∈ Z) is continuous.
Therefore, secant is continuous except at x = (2n + 1) π / 2 (n∈Z)
cot x = cos x / sin x, sin x ≠ 0 is continuous.
⇒ cot x, x ≠ nπ (n ∈ Z) is continuous.
Therefore, cotangent is continuous except at x = nπ (n ∈ Z)
NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 22
Discuss the continuity of the cosine, cosecant, secant, and cotangent functions
Summary:
Hence we can conclude that sin x is a continuous function. cos x is a continuous function. cosecant is continuous except at x = nπ (n ∈ Z). secant is continuous except at x = (2n + 1) π / 2 (n∈Z) .cotangent is continuous except at x = nπ (n ∈ Z)
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