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Question
Discuss the continuity of the cosine,cosecant,secant and cotangent functions.
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Solution
Checking continuity of cos x
Given function is f(x)=cos x
f(x)=cos x is trigonometric function. So, f(x) is continuous for all x ∈ R
Checking continuity of cosec x
Let g(x)=cosec x=1sin x
Let p(x)=1 Since, p(x) is a constant. so, p(x) is continuous.
Let q(x)=sin x is continuous for all real numbers. so, q(x) is contuinuous.
By algebra of continuous function
If p,q are continuous then pq is continuous when q≠0
Thus, g(x)=1sin x is continuous for all x ∈ R except the points where sin x=0
i.e., x=nπ,n ∈ Z
So, cosec x is continuous at all x ∈ R except the points where x=nπ,n ∈ Z
Checking continuity of sec x
Let h(x)=sec x=1cos x
Let p(x)=1. Since p(x) is a constant, So p(x) is continuous.
Let q(x)=cos x is continuous for all real numbers. so, q(x) is continuous.
By Algebra of continuous function
If p,q are continuous then pq is continuous when q≠0
Thus, h(x)=1cos x is continuous for all x ∈ R except the points where cos x=0
i.e., x=(2n+1)π2,n ∈ Z
So, sec x is continuous at all x ∈ R except the points where x=(2n+1)π2,n ∈ Z
Checking continuity of cot x
Let f(x)=cot x=cos xsin x
Let p(x)=cos x is continuous for all real numbers. so p(x) is continuous.
Let q(x)=sin x is continuous for all real numbers. so q(x) is continuous.
By Algebra of continuous function
Pf p,q are contonuous, then pq is continuous when q≠0.
Thus, f(x)=cos xsin x is contionuos for all x ∈ R except points where sin x=0
i.e., x=nπ,n ∈ Z
f(x)=cos x is continuous for all x ∈ R
g(x)=cosec x is continuous for all x ∈ R except where x=nπ,n ∈ Z
h(x)=sec x is continuous for all x ∈ R except where x=(2n+1)π2,n ∈ Z
j(x)=cot x is continuous for all x ∈ R except where x=nπ,n ∈ Z.
Given function is f(x)=cos x
f(x)=cos x is trigonometric function. So, f(x) is continuous for all x ∈ R
Checking continuity of cosec x
Let g(x)=cosec x=1sin x
Let p(x)=1 Since, p(x) is a constant. so, p(x) is continuous.
Let q(x)=sin x is continuous for all real numbers. so, q(x) is contuinuous.
By algebra of continuous function
If p,q are continuous then pq is continuous when q≠0
Thus, g(x)=1sin x is continuous for all x ∈ R except the points where sin x=0
i.e., x=nπ,n ∈ Z
So, cosec x is continuous at all x ∈ R except the points where x=nπ,n ∈ Z
Checking continuity of sec x
Let h(x)=sec x=1cos x
Let p(x)=1. Since p(x) is a constant, So p(x) is continuous.
Let q(x)=cos x is continuous for all real numbers. so, q(x) is continuous.
By Algebra of continuous function
If p,q are continuous then pq is continuous when q≠0
Thus, h(x)=1cos x is continuous for all x ∈ R except the points where cos x=0
i.e., x=(2n+1)π2,n ∈ Z
So, sec x is continuous at all x ∈ R except the points where x=(2n+1)π2,n ∈ Z
Checking continuity of cot x
Let f(x)=cot x=cos xsin x
Let p(x)=cos x is continuous for all real numbers. so p(x) is continuous.
Let q(x)=sin x is continuous for all real numbers. so q(x) is continuous.
By Algebra of continuous function
Pf p,q are contonuous, then pq is continuous when q≠0.
Thus, f(x)=cos xsin x is contionuos for all x ∈ R except points where sin x=0
i.e., x=nπ,n ∈ Z
f(x)=cos x is continuous for all x ∈ R
g(x)=cosec x is continuous for all x ∈ R except where x=nπ,n ∈ Z
h(x)=sec x is continuous for all x ∈ R except where x=(2n+1)π2,n ∈ Z
j(x)=cot x is continuous for all x ∈ R except where x=nπ,n ∈ Z.
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