Hsc maths formulae for board exam

1. HSC IMPORTANT FORMULAE FOR BOARD EXAM REFERENCE
1
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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
+ 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
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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
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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
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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