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

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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 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.