Law of Cosine (Cosine Law) - with Examples and Proof - Teachoo

Law of Cosine (Cosine Law) - Part 2
Law of Cosine (Cosine Law) - Part 3 Law of Cosine (Cosine Law) - Part 4 Law of Cosine (Cosine Law) - Part 5 Law of Cosine (Cosine Law) - Part 6 Law of Cosine (Cosine Law) - Part 7 Law of Cosine (Cosine Law) - Part 8 Law of Cosine (Cosine Law) - Part 9 Law of Cosine (Cosine Law) - Part 10 Law of Cosine (Cosine Law) - Part 11 Law of Cosine (Cosine Law) - Part 12 Law of Cosine (Cosine Law) - Part 13 Law of Cosine (Cosine Law) - Part 14

  1. Chapter 3 Class 11 Trigonometric Functions (Term 2)
  2. Concept wise

Transcript

In any βˆ†ABC, we have π‘Ž^2=𝑏^2+𝑐^2βˆ’2𝑏𝑐 cos⁑𝐴 or cos⁑𝐴=(𝑏^2 + 𝑐^2 βˆ’ π‘Ž^2)/2𝑏𝑐 𝑏^2=𝑐^2+π‘Ž^2βˆ’2π‘Žπ‘ cos⁑𝐡 or cos⁑𝐡=(π‘Ž^2 + 𝑐^2 βˆ’ 𝑏^2)/2π‘Žπ‘ 𝑐^2=π‘Ž^2+𝑏^2βˆ’2π‘Žπ‘ cos⁑𝐢 or cos⁑𝐢=(π‘Ž^2 + 𝑏^2 βˆ’ 𝑐^2)/2π‘Žπ‘ Proof of Cosine Rule There can be 3 cases - Acute Angled Triangle, Obtuse Angled Triangle, Right Angled Triangle Proof for Acute Angled Triangle Let’s draw perpendicular AD to BC In right triangle ABD cos B=𝐡𝐷/𝐴𝐡 cos B=𝐡𝐷/𝑐 𝑩𝐃=𝒄 𝒄𝒐𝒔⁑𝑩 In right triangle ACD cos C=𝐢𝐷/𝐴𝐢 cos C=𝐢𝐷/𝑏 π‘ͺ𝐃=𝒃 𝒄𝒐𝒔⁑π‘ͺ In βˆ†ACD, By Pythagoras theorem 〖𝐴𝐢〗^2=𝐴𝐷^2+𝐢𝐷^2 〖𝐴𝐢〗^2=𝐴𝐷^2+(π΅πΆβˆ’π΅π·)^2 〖𝐴𝐢〗^2=𝐴𝐷^2+𝐡𝐢^2+𝐡𝐷^2βˆ’2𝐡𝐢 . 𝐡𝐷 〖𝐴𝐢〗^2=𝐡𝐢^2+(𝑨𝑫^𝟐+𝑩𝑫^𝟐 )βˆ’2𝐡𝐢 . 𝐡𝐷 〖𝐴𝐢〗^2=𝐡𝐢^2+𝑨𝑩^πŸβˆ’2𝐡𝐢 . 𝐡𝐷 𝒃^𝟐=𝒂^𝟐+𝒄^πŸβˆ’πŸπ’‚π’„ γ€– 𝒄𝒐𝒔〗⁑𝑩 Similarly, we can prove others as well Proof for Obtuse Angled Triangle Let’s draw perpendicular AD to extended BC In right triangle ABD cos ∠ ABD=𝐡𝐷/𝐴𝐡 cos (180Β°βˆ’B)=𝐡𝐷/𝑐 βˆ’cos 𝐡=𝐡𝐷/𝑐 𝑩𝐃=βˆ’π’„ 𝒄𝒐𝒔⁑𝑩 In right triangle ACD cos C=𝐢𝐷/𝐴𝐢 cos C=𝐢𝐷/𝑏 𝐂𝑫=𝒃 𝒄𝒐𝒔⁑π‘ͺ In βˆ†ACD, By Pythagoras theorem 〖𝐴𝐢〗^2=𝐴𝐷^2+𝐢𝐷^2 〖𝐴𝐢〗^2=𝐴𝐷^2+(𝐡𝐢+𝐡𝐷)^2 〖𝐴𝐢〗^2=𝐴𝐷^2+𝐡𝐢^2+𝐡𝐷^2+2𝐡𝐢 . 𝐡𝐷 〖𝐴𝐢〗^2=𝐡𝐢^2+(𝑨𝑫^𝟐+𝑩𝑫^𝟐 )+2𝐡𝐢 . 𝐡𝐷 〖𝐴𝐢〗^2=𝐡𝐢^2+𝑨𝑩^𝟐+2𝐡𝐢 . 𝐡𝐷 𝑏^2=π‘Ž^2+𝑐^2+2 π‘Ž Γ— (βˆ’π‘ π‘π‘œπ‘ β‘π΅ ) 𝒃^𝟐=𝒂^𝟐+𝒄^πŸβˆ’πŸπ’‚π’„ 𝒄𝒐𝒔⁑𝑩 Similarly, we can prove others as well Proof for Right Angled TriangleSince ∠ B = 90Β° cos B = 0 In βˆ†ABC, By Pythagoras theorem 〖𝐴𝐢〗^2=𝐴𝐷^2+𝐢𝐢^2 𝑏^2=π‘Ž^2+𝑐^2 𝒃^𝟐=𝒂^𝟐+𝒄^πŸβˆ’πŸπ’‚π’„ 𝒄𝒐𝒔⁑𝑩 Similarly, we can prove others as well Let’s do some Examples!! Find the third side By Law of Cosines, π‘Ž^2=𝑏^2+𝑐^2βˆ’2𝑏𝑐 cos⁑𝐴 Putting values π‘Ž^2=9^2+12^2βˆ’2 Γ— 9 Γ— 12 Γ— cos⁑〖87Β°γ€— π‘Ž^2=81+144βˆ’216 Γ— 0.05 π‘Ž^2=225βˆ’11.3 π‘Ž=√213.7 π’‚β‰ˆπŸπŸ’.πŸ”πŸ One more Example ! Find the missing angle By Law of Cosines, 𝑏^2=π‘Ž^2+𝑏^2βˆ’2π‘Žπ‘ cos⁑𝐡 Putting values 20^2=60^2+50^2βˆ’2 Γ— 60 Γ— 50 Γ— cos⁑𝐡 400=3600+2500βˆ’6000 cos⁑𝐡 6000 cos⁑𝐡=6100βˆ’400 6000 cos⁑𝐡=5700 cos⁑𝐡=5700/6000 cos⁑𝐡=57/60 𝐡=cos^(βˆ’1)⁑〖19/20γ€— 𝐡=18.19Β° When to use Sine and Cosine Rule?Sine Rule is used when we are given 2 Angles and 1 Side (ASA) 2 Sides and 1 non-included angle (SSA) Cosine Rule is used when we are given 2 Sides and 1 included angle (SAS) 3 Sides (SSS)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.