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Hsc maths formulae for board exam
1.
HSC IMPORTANT FORMULAE
FOR 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 FORMULAE
FOR 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 FORMULAE
FOR 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 FORMULAE
FOR 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
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