hsc maths formulae for board exam

HSC IMPORTANT FORMULAE FOR BOARD EXAM REFERENCE

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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 + 3𝑎2𝑏 + 3𝑎𝑏2 + 𝑏3

6. 𝑎 − 𝑏 3 = 𝑎3 − 3𝑎2𝑏 + 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 = 𝑐𝑜𝑠𝑒𝑐2𝜃

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

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1. 𝑙𝑜𝑔 𝑎 + 𝑙𝑜𝑔 𝑏 = 𝑙𝑜𝑔 𝑎𝑏

2. 𝑙𝑜𝑔 𝑎 − 𝑙𝑜𝑔 𝑏 = 𝑙𝑜𝑔 𝑎

𝑏

3. 𝑙𝑜𝑔 𝑎𝑚 = 𝑚 𝑙𝑜𝑔 𝑎

4. 𝑎𝑙𝑜𝑔𝑎 𝑏 = 𝑏

5. 𝑙𝑜𝑔𝑎 𝑎 = 1

6. 𝑙𝑜𝑔𝑏 𝑎 =𝑙𝑜𝑔 𝑎

𝑙𝑜𝑔 𝑏

7. 𝑙𝑜𝑔 𝑒 = 1

8. 𝑙𝑜𝑔 1 = 0

1. 𝑎0 = 1

2. 𝑎𝑏 𝑚 = 𝑎𝑚𝑏𝑚

3. 𝑎

𝑏

𝑚

=𝑎𝑚

𝑏𝑚

4. 𝑎𝑚𝑎𝑛 = 𝑎𝑚+𝑛

5. 𝑎𝑚

𝑎𝑛 = 𝑎𝑚−𝑛

6. 𝑎−𝑚 =1

𝑎𝑚

7. 𝑎𝑚 =1

𝑎−𝑚

8. 𝑎𝑚 𝑛 = 𝑎𝑚𝑛

9. 𝑎 = 𝑎1

2

10. 𝑎3

= 𝑎1

3

11. 𝑎𝑛

= 𝑎1

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

𝑑𝑥𝑥𝑛 = 𝑛𝑥𝑛−1

2. 𝑑

𝑑𝑥𝑠𝑖𝑛 𝑥 = 𝑐𝑜𝑠 𝑥

3. 𝑑

𝑑𝑥𝑐𝑜𝑠 𝑥 = − 𝑠𝑖𝑛 𝑥

4. 𝑑

𝑑𝑥𝑡𝑎𝑛 𝑥 = 𝑠𝑒𝑐2 𝑥

5. 𝑑

𝑑𝑥𝑐𝑜𝑡 𝑥 = −𝑐𝑜𝑠𝑒𝑐2𝑥

6. 𝑑

𝑑𝑥𝑠𝑒𝑐 𝑥 = 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥

7. 𝑑

𝑑𝑥𝑐𝑜𝑠𝑒𝑐 𝑥 = −𝑐𝑜𝑠𝑒𝑐 𝑥 𝑐𝑜𝑡 𝑥

8. 𝑑

𝑑𝑥𝑙𝑜𝑔 𝑥 =

1

𝑥

9. 𝑑

𝑑𝑥𝑎𝑥 = 𝑎𝑥 𝑙𝑜𝑔 𝑎

10. 𝑑

𝑑𝑥 𝑘 = 0

11. 𝑑

𝑑𝑥𝑒𝑥 = 𝑒𝑥

12. 𝑑

𝑑𝑥 𝑥 = 1

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13. 𝑑

𝑑𝑥 𝑥 =

1

2 𝑥

14. 𝑑

𝑑𝑥𝑠𝑖𝑛−1 𝑥 =

1

1−𝑥2

15. 𝑑

𝑑𝑥𝑐𝑜𝑠−1 𝑥 =

−1

1−𝑥2

16. 𝑑

𝑑𝑥𝑡𝑎𝑛−1 𝑥 =

1

1+𝑥2

17. 𝑑

𝑑𝑥𝑐𝑜𝑡−1 𝑥 =

−1

1+𝑥2

18. 𝑑

𝑑𝑥𝑠𝑒𝑐−1 𝑥 =

1

𝑥 𝑥2−1

19. 𝑑

𝑑𝑥𝑐𝑜𝑠𝑒𝑐−1 𝑥 =

−1

𝑥 𝑥2−1

Integration formulae

1. 0 𝑑𝑥 = 𝐶 𝑤ℎ𝑒𝑟𝑒 𝑐 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

2. 1 𝑑𝑥 = 𝑥 + 𝑐

3. 𝑥𝑛𝑑𝑥 =𝑥𝑛+1

𝑛+1+ 𝑐

4. 1

𝑥𝑑𝑥 = 2 𝑥 + 𝑐

5. 1

𝑥𝑑𝑥 = 𝑙𝑜𝑔𝑥 + 𝑐

6. 𝑒𝑥𝑑𝑥 = 𝑒𝑥 + 𝑐

7. 𝑎𝑥𝑑𝑥 =𝑎𝑥

𝑙𝑜𝑔𝑎+ 𝑐

8. 𝑠𝑖𝑛𝑥 𝑑𝑥 = −𝑐𝑜𝑠𝑥 + 𝑐

9. 𝑐𝑜𝑠𝑥 𝑑𝑥 = 𝑠𝑖𝑛𝑥 + 𝑐

10. 𝑠𝑒𝑐2 𝑥𝑑𝑥 = 𝑡𝑎𝑛𝑥 + 𝑐

11. 𝑐𝑜𝑠𝑒𝑐2𝑥 𝑑𝑥 = −𝑐𝑜𝑡𝑥 + 𝑐

12. 𝑠𝑒𝑐𝑥 𝑡𝑎𝑛𝑥 𝑑𝑥 = 𝑠𝑒𝑐𝑥 + 𝑐

13. 𝑐𝑜𝑠𝑒𝑐𝑥 𝑐𝑜𝑡𝑥 𝑑𝑥 = −𝑐𝑜𝑠𝑒𝑐𝑥 + 𝑐

14. 1

1−𝑥2𝑑𝑥 = 𝑠𝑖𝑛−1 𝑥 + 𝑐 /−𝑐𝑜𝑠−1 𝑥 + 𝑐

15. 1

1+𝑥2 𝑑𝑥 = 𝑡𝑎𝑛−1 𝑥 + 𝑐 / −𝑐𝑜𝑡−1 𝑥 + 𝑐

16. 1

𝑥 𝑥2−1𝑑𝑥 = 𝑠𝑒𝑐−1 𝑥 + 𝑐 /−𝑐𝑜𝑠𝑒𝑐−1𝑥 + 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+𝑎2𝑑𝑥 = 𝑙𝑜𝑔 𝑥 + 𝑥2 + 𝑎2 + 𝑐

22. 1

𝑥2−𝑎2𝑑𝑥 = 𝑙𝑜𝑔 𝑥 + 𝑥2 − 𝑎2 + 𝑐

23. 1

𝑎2−𝑥2𝑑𝑥 = 𝑠𝑖𝑛−1

𝑥

𝑎 + 𝑐

24. 1

𝑥2+𝑎2 𝑑𝑥 =1

𝑎𝑡𝑎𝑛−1

𝑥

𝑎 + 𝑐

25. 1

𝑥2−𝑎2 𝑑𝑥 =1

2𝑎𝑙𝑜𝑔

𝑥−𝑎

𝑥+𝑎 + 𝑐

26. 1

𝑎2−𝑥2 𝑑𝑥 =1

2𝑎𝑙𝑜𝑔

𝑎+𝑥

𝑎−𝑥 + 𝑐

27. 1

𝑥 𝑥2−𝑎2𝑑𝑥 =

1

𝑎𝑠𝑒𝑐−1

𝑥

𝑎 + 𝑐

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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