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Discuss the continuity of cosine function [with Video] - Teachoo

Ex 5.1, 22 (i) - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.1, 22 (i) - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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Ex 5.1, 22 (i) Discuss the continuity of the cosine, cosecant, secant and cotangent functions.Let 𝒇(𝒙)=πœπ¨π¬β‘π’™ To check continuity of 𝑓(π‘₯), We check it’s if it is continuous at any point x = c Let c be any real number f is continuous at π‘₯ =𝑐 if if L.H.L = R.H.L = 𝑓(𝑐) i.e. lim┬(x→𝑐^βˆ’ ) 𝑓(π‘₯)= lim┬(x→𝑐^+ ) " " 𝑓(π‘₯)= 𝑓(𝑐) LHL at x β†’ c lim┬(x→𝑐^βˆ’ ) f(x) = lim┬(hβ†’0) f(c βˆ’ h) = lim┬(hβ†’0) cos⁑(π‘βˆ’β„Ž) = lim┬(hβ†’0) sin⁑𝑐 sinβ‘β„Ž+cos⁑𝑐 cosβ‘β„Ž Putting β„Ž=0 = sin⁑𝑐 sin⁑0+cos⁑𝑐 cos⁑0 = 0+cos c . 1 = 𝒄𝒐𝒔⁑𝒄 𝐴𝑠, cos⁑(π‘₯βˆ’π‘¦) =cos⁑π‘₯ cosβ‘π‘¦βˆ’sin⁑π‘₯ sin⁑𝑦 RHL at x β†’ c lim┬(x→𝑐^+ ) f(x) = lim┬(hβ†’0) f(c + h) = lim┬(hβ†’0) cos⁑(𝑐+β„Ž) = lim┬(hβ†’0) cos⁑𝑐 cosβ‘β„Ž – sin⁑𝑐 sinβ‘β„Ž Putting β„Ž=0 = cos⁑𝑐 cos⁑0 – sin⁑𝑐 sin⁑0 = cos c . 1 – 0 = 𝒄𝒐𝒔⁑𝒄 𝐴𝑠, cos⁑(π‘₯+𝑦) =cos⁑π‘₯ cosβ‘π‘¦βˆ’sin⁑π‘₯ sin⁑𝑦 And, 𝑓(𝑐) = cos⁑𝑐 Since, L.H.L = R.H.L = 𝑓(𝑐) ∴ Function is continuous at x = c Thus, we can write that f is continuous for x = c , where c βˆˆπ‘ ∴ 𝒄𝒐𝒔⁑𝒙 is continuous for every real number.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.