web analytics

Misc 15 - Using properties of determinants, prove |sin cos - Whole row/column zero

Slide69.JPG
Slide70.JPG

  1. Class 12
  2. Important Question for exams Class 12
Ask Download

Transcript

Misc. 15 Using properties of determinants, prove that: sin﷮α﷯﷮ cos﷮α﷯﷮ cos﷮(α+δ)﷯﷮ sin﷮β﷯﷮ cos﷮β﷯﷮ cos﷮(β+δ)﷯﷮ sin﷮γ﷯﷮ cos﷮γ﷯﷮ cos﷮(γ+δ)﷯﷯﷯ = 0 Let ∆ = sin﷮α﷯﷮ cos﷮α﷯﷮ cos﷮(α+δ)﷯﷮ sin﷮β﷯﷮ cos﷮β﷯﷮ cos﷮(β+δ)﷯﷮ sin﷮γ﷯﷮ cos﷮γ﷯﷮ cos﷮(γ+δ)﷯﷯﷯ Using cos (x + y) = cos x cos y – sin x sin y = sin﷮α﷯﷮ cos﷮α﷯﷮cos𝛼 cos﷮δ − sin﷮𝛼 sin﷮𝛿﷯﷯﷯﷮ sin﷮β﷯﷮ cos﷮β﷯﷮ cos﷮𝛽﷯ cos﷮𝛿− sin﷮𝛽 sin﷮𝛿﷯﷯﷯﷮ sin﷮γ﷯﷮ cos﷮γ﷯﷮cosγcos 𝛿 − sin﷮γ sin﷮𝛿﷯﷯﷯﷯ Expressing elements of 2nd row as sum of two elements Using Property : If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms ,then the determinant is expressed as a sum of two (or more) determinants. = sin﷮α﷯﷮ cos﷮α﷯﷮cos 𝛼 cos﷮δ ﷯﷮ sin﷮β﷯﷮ cos﷮β﷯﷮ cos﷮𝛽﷯ cos﷮𝛿﷯﷮ sin﷮γ﷯﷮ cos﷮γ﷯﷮cos γ cos 𝛿 ﷯﷯ + sin﷮α﷯﷮ cos﷮α﷯﷮− sin﷮𝛼 sin﷮𝛿﷯﷯﷮ sin﷮β﷯﷮ cos﷮β﷯﷮− sin﷮𝛽 sin﷮𝛿﷯﷯﷮ sin﷮γ﷯﷮ cos﷮γ﷯﷮− sin﷮γ sin﷮𝛿﷯﷯﷯﷯ = cos﷮δ ﷯ sin﷮α﷯﷮ cos﷮α﷯﷮cos 𝛼﷮ sin﷮β﷯﷮ cos﷮β﷯﷮ cos﷮𝛽﷯﷮ sin﷮γ﷯﷮ cos﷮γ﷯﷮cos γ ﷯﷯ + (− sin﷮𝛿﷯) sin﷮α﷯﷮ cos﷮α﷯﷮ sin﷮𝛼﷯﷮ sin﷮β﷯﷮ cos﷮β﷯﷮ sin﷮𝛽﷯﷮ sin﷮γ﷯﷮ cos﷮γ﷯﷮ sin﷮γ﷯﷯﷯ = cos﷮δ ﷯(0) + (− sin﷮δ﷯) (0) = 0 = R.H.S Hence proved

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
Jail