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Ex 3.2, 13 - Show that F(x) F(y) = F(x + y), If F(x) = [cos x

Ex 3.2, 13 - Chapter 3 Class 12 Matrices - Part 2
Ex 3.2, 13 - Chapter 3 Class 12 Matrices - Part 3 Ex 3.2, 13 - Chapter 3 Class 12 Matrices - Part 4


Transcript

Ex 3.2, 13 If F (x) = [β– 8(cos⁑π‘₯&γ€–βˆ’sin〗⁑π‘₯&[email protected]⁑π‘₯&cos⁑π‘₯&[email protected]&0&1)] , Show that F(x) F(y) = F(x + y) We need to show F(x) F(y) = F(x + y) Taking L.H.S. Given F(x) = [β– 8(cos⁑π‘₯&γ€–βˆ’sin〗⁑π‘₯&[email protected]⁑π‘₯&cos⁑π‘₯&[email protected]&0&1)] Finding F(y) Replacing x by y in F(x) F(y) = [β– 8(cos⁑𝑦&γ€–βˆ’sin〗⁑𝑦&[email protected]⁑𝑦&cos⁑𝑦&[email protected]&0&1)] Now, F(x) F(y) = [β– 8(cos⁑π‘₯&γ€–βˆ’sin〗⁑π‘₯&[email protected]⁑π‘₯&cos⁑π‘₯&[email protected]&0&1)] [β– 8(cos⁑𝑦&γ€–βˆ’sin〗⁑𝑦&[email protected]⁑𝑦&cos⁑𝑦&[email protected]&0&1)] = [β– 8(cos⁑π‘₯ cos⁑𝑦+(γ€–βˆ’sin〗⁑π‘₯ ) sin⁑〖𝑦+0 γ€— &cos⁑〖π‘₯(βˆ’sin⁑〖𝑦)+(βˆ’sin⁑〖π‘₯)γ€–cos y〗⁑〖+ 0γ€— γ€— γ€— γ€—&0+0+0Γ—[email protected]⁑〖π‘₯ cos⁑〖𝑦+cos⁑〖π‘₯ sin⁑〖𝑦+0γ€— γ€— γ€— γ€—&sin⁑π‘₯ (βˆ’sin⁑〖𝑦)+γ€— cos⁑〖π‘₯ cos⁑〖𝑦+0γ€— γ€—&0+0+0Γ—[email protected]Γ—cos⁑〖𝑦 +0Γ—sin⁑〖𝑦+0Γ—1γ€— γ€—&0Γ—(βˆ’sin⁑〖𝑦)+0Γ—cos⁑〖𝑦+0γ€— γ€—&0+0+1Γ—1)] = [β– 8(cos⁑π‘₯ cos⁑𝑦 γ€–βˆ’sin〗⁑π‘₯.sin⁑〖𝑦 γ€— &γ€–βˆ’cos〗⁑〖π‘₯ sinβ‘γ€–π‘¦βˆ’sin⁑〖π‘₯ cos⁑𝑦 γ€— γ€— γ€—&[email protected]⁑〖π‘₯ cos⁑〖𝑦+cos⁑〖π‘₯ sin⁑𝑦 γ€— γ€— γ€—&βˆ’sin⁑π‘₯ sin⁑〖𝑦+γ€— cos⁑〖π‘₯ cos⁑𝑦 γ€—&[email protected]&0&1)] We know that cos x cos y – sin x sin y = cos (x + y) & sin x cos y + cos x sin y = sin (x + y) = [β– 8(cos⁑〖(π‘₯+𝑦)γ€— &γ€–βˆ’[cos〗⁑〖π‘₯ sin⁑〖𝑦+sin⁑〖π‘₯ cos⁑〖𝑦]γ€— γ€— γ€— γ€—&[email protected]⁑〖(π‘₯+𝑦)γ€—&cos⁑π‘₯ cos⁑〖𝑦 βˆ’γ€— sin⁑〖π‘₯ sin⁑𝑦 γ€—&[email protected]&0&1)] = [β– 8(cos⁑〖(π‘₯+𝑦)γ€— &βˆ’sin⁑〖(π‘₯+𝑦)γ€—&[email protected]⁑〖(π‘₯+𝑦)γ€—&cos⁑〖(π‘₯+𝑦)γ€—&[email protected]&0&1)] Taking R.H.S F(x + y) Replacing x by (x + y) in F(x) = [β– 8(cos⁑〖(π‘₯+𝑦)γ€— &βˆ’sin⁑〖(π‘₯+𝑦)γ€—&[email protected]⁑〖(π‘₯+𝑦)γ€—&cos⁑〖(π‘₯+𝑦)γ€—&[email protected]&0&1)] = L.H.S. Hence proved

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.