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Hsc maths formulae for board exam
This document provides a comprehensive reference of important algebraic, trigonometric, logarithmic, and derivative formulae for the Higher Secondary Certificate (HSC) board exam. It includes over 100 formulae across various mathematical domains to assist exam preparation. Key formulae covered include trigonometric identities, logarithmic properties, derivatives of basic functions, and algebraic manipulations of exponents and radicals.
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Hsc maths formulae for board exam
- 1. HSC IMPORTANT FORMULAEFOR BOARD EXAM REFERENCE 1 www.omtexclasses.com e-mail: student@omtexclasses.com ALGEBRAIC FORMULAE 1. 𝑎0 = 1 2. 𝑎 + 𝑏 2 = 𝑎2 + 2𝑎𝑏 + 𝑏2 3. 𝑎 − 𝑏 2 = 𝑎2 − 2𝑎𝑏 + 𝑏2 4. 𝑎2 − 𝑏2 = 𝑎 + 𝑏 (𝑎 − 𝑏) 5. 𝑎 + 𝑏 3 = 𝑎3 + 3𝑎2 𝑏 + 3𝑎𝑏2 + 𝑏3 6. 𝑎 − 𝑏 3 = 𝑎3 − 3𝑎2 𝑏 + 3𝑎𝑏2 − 𝑏3 7. 𝑎3 + 𝑏3 = 𝑎 + 𝑏 𝑎2 − 𝑎𝑏 + 𝑏2 8. 𝑎3 − 𝑏3 = 𝑎 − 𝑏 𝑎2 + 𝑎𝑏 + 𝑏2 TRIGONOMETRIC FORMULAE 1. 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠2 𝜃 = 1 2. 1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐2 𝜃 3. 1 + 𝑐𝑜𝑡2 𝜃 = 𝑐𝑜𝑠𝑒𝑐2 𝜃 4. 𝑡𝑎𝑛𝜃 = 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 = 1 𝑐𝑜𝑡𝜃 5. 𝑐𝑜𝑡𝜃 = 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 = 1 𝑡𝑎𝑛𝜃 6. 𝑠𝑒𝑐𝜃 = 1 𝑐𝑜𝑠𝜃 7. 𝑐𝑜𝑠𝑒𝑐𝜃 = 1 𝑠𝑖𝑛𝜃 8. 𝑠𝑖𝑛2𝜃 = 2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 = 2𝑡𝑎𝑛𝜃 (1+𝑡𝑎𝑛 2 𝜃) 9. 𝑐𝑜𝑠2𝜃 = 𝑐𝑜𝑠2 𝜃 − 𝑠𝑖𝑛2 𝜃 = 2 𝑐𝑜𝑠2 𝜃 − 1 = 1 − 2 𝑠𝑖𝑛2 𝜃 = 1−𝑡𝑎𝑛 2 𝜃 1+𝑡𝑎𝑛 2 𝜃 10. 𝑡𝑎𝑛2𝜃 = 2𝑡𝑎𝑛𝜃 1−𝑡𝑎𝑛 2 𝜃 11. 𝑠𝑖𝑛3𝜃 = 3𝑠𝑖𝑛𝜃 − 4 𝑠𝑖𝑛3 𝜃 12. 𝑐𝑜𝑠3𝜃 = 4 𝑐𝑜𝑠3 𝜃 − 3𝑐𝑜𝑠𝜃 13. 𝑡𝑎𝑛3𝜃 = 3𝑡𝑎𝑛𝜃 −𝑡𝑎𝑛 3 𝜃 1−3 𝑡𝑎𝑛 2 𝜃 14. 1 + 𝑐𝑜𝑠2𝜃 = 2 𝑐𝑜𝑠2 𝜃 15. 1 − 𝑐𝑜𝑠2𝜃 = 2 𝑠𝑖𝑛2 𝜃 16. 1 + 𝑠𝑖𝑛2𝜃 = 𝑐𝑜𝑠𝜃 + 𝑠𝑖𝑛𝜃 2 17. 1 − 𝑠𝑖𝑛2𝜃 = 𝑐𝑜𝑠𝜃 − 𝑠𝑖𝑛𝜃 2 18. 𝑠𝑖𝑛 𝑎 + 𝑏 = 𝑠𝑖𝑛 𝑎 𝑐𝑜𝑠 𝑏 + 𝑐𝑜𝑠 𝑎 𝑠𝑖𝑛 𝑏 19. 𝑠𝑖𝑛 𝑎 − 𝑏 = 𝑠𝑖𝑛 𝑎 𝑐𝑜𝑠 𝑏 − 𝑐𝑜𝑠 𝑎 𝑠𝑖𝑛 𝑏 20. 𝑐𝑜𝑠 𝑎 + 𝑏 = 𝑐𝑜𝑠 𝑎 𝑐𝑜𝑠 𝑏 − 𝑠𝑖𝑛 𝑎 𝑠𝑖𝑛 𝑏 21. 𝑐𝑜𝑠 𝑎 − 𝑏 = 𝑐𝑜𝑠 𝑎 𝑐𝑜𝑠 𝑏 + 𝑠𝑖𝑛 𝑎 𝑠𝑖𝑛 𝑏 1. 𝑠𝑖𝑛 𝐴 + 𝑠𝑖𝑛 𝐵 = 2 𝑠𝑖𝑛 𝐴+𝐵 2 𝑐𝑜𝑠 𝐴−𝐵 2 2. 𝑠𝑖𝑛 𝐴 − 𝑠𝑖𝑛 𝐵 = 2 𝑐𝑜𝑠 𝐴+𝐵 2 𝑠𝑖𝑛 𝐴−𝐵 2 3. 𝑐𝑜𝑠 𝐴 + 𝑐𝑜𝑠 𝐵 = 2 𝑐𝑜𝑠 𝐴+𝐵 2 𝑐𝑜𝑠 𝐴−𝐵 2 4. 𝑐𝑜𝑠 𝐴 − 𝑐𝑜𝑠 𝐵 = −2 𝑠𝑖𝑛 𝐴+𝐵 2 𝑠𝑖𝑛 𝐴−𝐵 2 5. 𝑠𝑖𝑛𝐴 𝑐𝑜𝑠𝐵 = 1 2 𝑠𝑖𝑛 𝐴 + 𝐵 + 𝑠𝑖𝑛 𝐴 − 𝐵 6. 𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐵 = 1 2 𝑠𝑖𝑛 𝐴 + 𝐵 − 𝑠𝑖𝑛 𝐴 − 𝐵 7. 𝑐𝑜𝑠𝐴 𝑐𝑜𝑠𝐵 = 1 2 𝑐𝑜𝑠 𝐴 + 𝐵 + 𝑐𝑜𝑠 𝐴 − 𝐵 8. 𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵 = − 1 2 𝑐𝑜𝑠 𝐴 + 𝐵 − 𝑐𝑜𝑠 𝐴 − 𝐵 1. 𝑡𝑎𝑛 𝐴 + 𝐵 = 𝑡𝑎𝑛𝐴 +𝑡𝑎𝑛𝐵 1−𝑡𝑎𝑛𝐴𝑡𝑎𝑛𝐵 2. 𝑡𝑎𝑛 𝐴 − 𝐵 = 𝑡𝑎𝑛𝐴 −𝑡𝑎𝑛𝐵 1+𝑡𝑎𝑛𝐴𝑡𝑎𝑛𝐵 3. 𝑡𝑎𝑛 𝜋 4 + 𝜃 = 1+𝑡𝑎𝑛𝜃 1−𝑡𝑎𝑛𝜃 4. 𝑡𝑎𝑛 𝜋 4 − 𝜃 = 1−𝑡𝑎𝑛𝜃 1+𝑡𝑎𝑛𝜃 5. 𝑡𝑎𝑛−1 𝑥+𝑦 1−𝑥𝑦 = 𝑡𝑎𝑛−1 𝑥 + 𝑡𝑎𝑛−1 𝑦 6. 𝑡𝑎𝑛−1 𝑥−𝑦 1+𝑥𝑦 = 𝑡𝑎𝑛−1 𝑥 − 𝑡𝑎𝑛−1 𝑦
- 2. HSC IMPORTANT FORMULAEFOR BOARD EXAM REFERENCE 2 www.omtexclasses.com e-mail: student@omtexclasses.com 1. 𝑙𝑜𝑔 𝑎 + 𝑙𝑜𝑔 𝑏 = 𝑙𝑜𝑔 𝑎𝑏 2. 𝑙𝑜𝑔 𝑎 − 𝑙𝑜𝑔 𝑏 = 𝑙𝑜𝑔 𝑎 𝑏 3. 𝑙𝑜𝑔 𝑎 𝑚 = 𝑚 𝑙𝑜𝑔 𝑎 4. 𝑎𝑙𝑜𝑔 𝑎 𝑏 = 𝑏 5. 𝑙𝑜𝑔 𝑎 𝑎 = 1 6. 𝑙𝑜𝑔 𝑏 𝑎 = 𝑙𝑜𝑔 𝑎 𝑙𝑜𝑔 𝑏 7. 𝑙𝑜𝑔 𝑒 = 1 8. 𝑙𝑜𝑔 1 = 0 1. 𝑎0 = 1 2. 𝑎𝑏 𝑚 = 𝑎 𝑚 𝑏 𝑚 3. 𝑎 𝑏 𝑚 = 𝑎 𝑚 𝑏 𝑚 4. 𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚+𝑛 5. 𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚−𝑛 6. 𝑎−𝑚 = 1 𝑎 𝑚 7. 𝑎 𝑚 = 1 𝑎−𝑚 8. 𝑎 𝑚 𝑛 = 𝑎 𝑚𝑛 9. 𝑎 = 𝑎 1 2 10. 𝑎 3 = 𝑎 1 3 11. 𝑎 𝑛 = 𝑎 1 𝑛 1. 𝑠𝑖𝑛 −𝜃 = − 𝑠𝑖𝑛 𝜃 2. 𝑐𝑜𝑠 −𝜃 = 𝑐𝑜𝑠 𝜃 3. 𝑡𝑎𝑛 −𝜃 = −𝑡𝑎𝑛 𝜃 4. 𝑠𝑖𝑛 𝜋 2 − 𝜃 = 𝑐𝑜𝑠 𝜃 5. 𝑐𝑜𝑠 𝜋 2 − 𝜃 = 𝑠𝑖𝑛 𝜃 6. 𝑡𝑎𝑛 𝜋 2 − 𝜃 = 𝑐𝑜𝑡 𝜃 7. 𝑐𝑜𝑡 𝜋 2 − 𝜃 = 𝑡𝑎𝑛 𝜃 8. 𝑐𝑜𝑠𝑒𝑐 𝜋 2 − 𝜃 = 𝑠𝑒𝑐 𝜃 9. 𝑠𝑒𝑐 𝜋 2 − 𝜃 = 𝑐𝑜𝑠𝑒𝑐 𝜃 10. 𝑠𝑖𝑛 𝜋 2 + 𝜃 = 𝑐𝑜𝑠 𝜃 11. 𝑐𝑜𝑠 𝜋 2 + 𝜃 = − 𝑠𝑖𝑛 𝜃 12. 𝑠𝑖𝑛 𝜋 + 𝜃 = − 𝑠𝑖𝑛 𝜃 13. 𝑐𝑜𝑠 𝜋 + 𝜃 = − 𝑐𝑜𝑠 𝜃 14. 𝑠𝑖𝑛 𝜋 − 𝜃 = 𝑠𝑖𝑛 𝜃 15. 𝑐𝑜𝑠 𝜋 − 𝜃 = − 𝑐𝑜𝑠 𝜃 DERIVATIVES FORMULAE 1. 𝑑 𝑑𝑥 𝑥 𝑛 = 𝑛𝑥 𝑛−1 2. 𝑑 𝑑𝑥 𝑠𝑖𝑛 𝑥 = 𝑐𝑜𝑠 𝑥 3. 𝑑 𝑑𝑥 𝑐𝑜𝑠 𝑥 = − 𝑠𝑖𝑛 𝑥 4. 𝑑 𝑑𝑥 𝑡𝑎𝑛 𝑥 = 𝑠𝑒𝑐2 𝑥 5. 𝑑 𝑑𝑥 𝑐𝑜𝑡 𝑥 = −𝑐𝑜𝑠𝑒𝑐2 𝑥 6. 𝑑 𝑑𝑥 𝑠𝑒𝑐 𝑥 = 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥 7. 𝑑 𝑑𝑥 𝑐𝑜𝑠𝑒𝑐 𝑥 = −𝑐𝑜𝑠𝑒𝑐 𝑥 𝑐𝑜𝑡 𝑥 8. 𝑑 𝑑𝑥 𝑙𝑜𝑔 𝑥 = 1 𝑥 9. 𝑑 𝑑𝑥 𝑎 𝑥 = 𝑎 𝑥 𝑙𝑜𝑔 𝑎 10. 𝑑 𝑑𝑥 𝑘 = 0 11. 𝑑 𝑑𝑥 𝑒 𝑥 = 𝑒 𝑥 12. 𝑑 𝑑𝑥 𝑥 = 1
- 3. HSC IMPORTANT FORMULAEFOR BOARD EXAM REFERENCE 3 www.omtexclasses.com e-mail: student@omtexclasses.com 13. 𝑑 𝑑𝑥 𝑥 = 1 2 𝑥 14. 𝑑 𝑑𝑥 𝑠𝑖𝑛−1 𝑥 = 1 1−𝑥2 15. 𝑑 𝑑𝑥 𝑐𝑜𝑠−1 𝑥 = −1 1−𝑥2 16. 𝑑 𝑑𝑥 𝑡𝑎𝑛−1 𝑥 = 1 1+𝑥2 17. 𝑑 𝑑𝑥 𝑐𝑜𝑡−1 𝑥 = −1 1+𝑥2 18. 𝑑 𝑑𝑥 𝑠𝑒𝑐−1 𝑥 = 1 𝑥 𝑥2−1 19. 𝑑 𝑑𝑥 𝑐𝑜𝑠𝑒𝑐−1 𝑥 = −1 𝑥 𝑥2−1 Integration formulae 1. 0 𝑑𝑥 = 𝐶 𝑤ℎ𝑒𝑟𝑒 𝑐 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 2. 1 𝑑𝑥 = 𝑥 + 𝑐 3. 𝑥 𝑛 𝑑𝑥 = 𝑥 𝑛+1 𝑛+1 + 𝑐 4. 1 𝑥 𝑑𝑥 = 2 𝑥 + 𝑐 5. 1 𝑥 𝑑𝑥 = 𝑙𝑜𝑔𝑥 + 𝑐 6. 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝑐 7. 𝑎 𝑥 𝑑𝑥 = 𝑎 𝑥 𝑙𝑜𝑔𝑎 + 𝑐 8. 𝑠𝑖𝑛𝑥 𝑑𝑥 = −𝑐𝑜𝑠𝑥 + 𝑐 9. 𝑐𝑜𝑠𝑥 𝑑𝑥 = 𝑠𝑖𝑛𝑥 + 𝑐 10. 𝑠𝑒𝑐2 𝑥𝑑𝑥 = 𝑡𝑎𝑛𝑥 + 𝑐 11. 𝑐𝑜𝑠𝑒𝑐2 𝑥 𝑑𝑥 = −𝑐𝑜𝑡𝑥 + 𝑐 12. 𝑠𝑒𝑐𝑥 𝑡𝑎𝑛𝑥 𝑑𝑥 = 𝑠𝑒𝑐𝑥 + 𝑐 13. 𝑐𝑜𝑠𝑒𝑐𝑥 𝑐𝑜𝑡𝑥 𝑑𝑥 = −𝑐𝑜𝑠𝑒𝑐𝑥 + 𝑐 14. 1 1−𝑥2 𝑑𝑥 = 𝑠𝑖𝑛−1 𝑥 + 𝑐 /− 𝑐𝑜𝑠−1 𝑥 + 𝑐 15. 1 1+𝑥2 𝑑𝑥 = 𝑡𝑎𝑛−1 𝑥 + 𝑐 / − 𝑐𝑜𝑡−1 𝑥 + 𝑐 16. 1 𝑥 𝑥2−1 𝑑𝑥 = 𝑠𝑒𝑐−1 𝑥 + 𝑐 /−𝑐𝑜𝑠𝑒𝑐−1 𝑥 + 𝑐 17. 𝑡𝑎𝑛𝑥 𝑑𝑥 = 𝑙𝑜𝑔 𝑠𝑒𝑐𝑥 + 𝑐 18. 𝑐𝑜𝑡𝑥 𝑑𝑥 = 𝑙𝑜𝑔 𝑠𝑖𝑛 𝑥 + 𝑐 19. 𝑠𝑒𝑐𝑥 𝑑𝑥 = 𝑙𝑜𝑔 𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥 + 𝑐 20. 𝑐𝑜𝑠𝑒𝑐𝑥 𝑑𝑥 = 𝑙𝑜𝑔 𝑐𝑜𝑠𝑒𝑐𝑥 − 𝑐𝑜𝑡𝑥 + 𝑐 21. 1 𝑥2+𝑎2 𝑑𝑥 = 𝑙𝑜𝑔 𝑥 + 𝑥2 + 𝑎2 + 𝑐 22. 1 𝑥2−𝑎2 𝑑𝑥 = 𝑙𝑜𝑔 𝑥 + 𝑥2 − 𝑎2 + 𝑐 23. 1 𝑎2−𝑥2 𝑑𝑥 = 𝑠𝑖𝑛−1 𝑥 𝑎 + 𝑐 24. 1 𝑥2+𝑎2 𝑑𝑥 = 1 𝑎 𝑡𝑎𝑛−1 𝑥 𝑎 + 𝑐 25. 1 𝑥2−𝑎2 𝑑𝑥 = 1 2𝑎 𝑙𝑜𝑔 𝑥−𝑎 𝑥+𝑎 + 𝑐 26. 1 𝑎2−𝑥2 𝑑𝑥 = 1 2𝑎 𝑙𝑜𝑔 𝑎+𝑥 𝑎−𝑥 + 𝑐 27. 1 𝑥 𝑥2−𝑎2 𝑑𝑥 = 1 𝑎 𝑠𝑒𝑐−1 𝑥 𝑎 + 𝑐
- 4. HSC IMPORTANT FORMULAEFOR BOARD EXAM REFERENCE 4 www.omtexclasses.com e-mail: student@omtexclasses.com 𝜃 00 00 300 𝜋 6 450 𝜋 4 600 𝜋 3 900 𝜋 2 1800 𝜋 2700 3𝜋 2 3600 2𝜋 sin 𝜃 0 1 2 1 2 3 2 1 0 −1 0 cos 𝜃 1 3 2 1 2 1 2 0 −1 0 1 tan 𝜃 0 1 3 1 3 𝑁. 𝐷. 0 𝑁. 𝐷. 0 cosec 𝜃 𝑁. 𝐷. 2 2 2 3 1 𝑁. 𝐷. −1 𝑁. 𝐷 sec 𝜃 1 2 3 2 2 𝑁. 𝐷. −1 𝑁. 𝐷. 1 cot 𝜃 𝑁. 𝐷. 3 1 1 3 0 𝑁. 𝐷. 0 𝑁. 𝐷