Mathematics Formulae for F5 Students
Trigonometry formulae
Basic trigonometry formulae
General solution
𝑠𝑖𝑛2 𝐴 + 𝑐𝑜𝑠 2 𝐴 = 1
𝑠𝑖𝑛 𝐴 → 180𝑛° + (−1)𝑛 𝐴
or 𝑛𝜋 + (−1)𝑛 𝐴
𝑡𝑎𝑛2 𝐴 + 1 = 𝑠𝑒𝑐 2 𝐴
𝑐𝑜𝑠 𝐴 → 360𝑛° ± 𝐴
or 2𝑛𝜋 ± 𝐴
𝑐𝑜𝑡 2 𝐴 + 1 = 𝑐𝑠𝑐 2 𝐴
𝑡𝑎𝑛 𝐴 → 180𝑛° + 𝐴
or 𝑛𝜋 + 𝐴
Half angle formulae
Double angle formulae
𝑠𝑖𝑛2
𝜃 1
= (1 − 𝑐𝑜𝑠 𝜃)
2 2
𝜃 1
𝑐𝑜𝑠
= (1 + 𝑐𝑜𝑠 𝜃)
2 2
𝑠𝑖𝑛 2𝐴 = 2 𝑠𝑖𝑛 𝐴 𝑐𝑜𝑠 𝐴
𝑐𝑜𝑠 2𝐴 = 𝑐𝑜𝑠 2 𝐴 − 𝑠𝑖𝑛2 𝐴
2
𝑡𝑎𝑛
𝜃 1 − 𝑐𝑜𝑠 𝜃
=
2
𝑠𝑖𝑛 𝜃
= 2𝑐𝑜𝑠 2 𝐴 − 1
= 1 − 2𝑠𝑖𝑛2 𝐴
𝑡𝑎𝑛 2𝐴 =
2 𝑡𝑎𝑛 𝐴
1 − 𝑡𝑎𝑛2 𝐴
Triple angle formulae
Compound Angle
𝑠𝑖𝑛 3𝐴 = 3 𝑠𝑖𝑛 𝐴 − 4𝑠𝑖𝑛3 𝐴
𝑠𝑖𝑛(𝐴 + 𝐵) = 𝑠𝑖𝑛 𝐴 𝑐𝑜𝑠 𝐵 + 𝑐𝑜𝑠 𝐴 𝑠𝑖𝑛 𝐵
𝑐𝑜𝑠 3𝐴 = 4 𝑐𝑜𝑠 3 𝐴 − 3 𝑐𝑜𝑠 𝐴
𝑠𝑖𝑛(𝐴 − 𝐵) = 𝑠𝑖𝑛 𝐴 𝑐𝑜𝑠 𝐵 − 𝑐𝑜𝑠 𝐴 𝑠𝑖𝑛 𝐵
3 𝑡𝑎𝑛 𝐴 − 𝑡𝑎𝑛3 𝐴
𝑡𝑎𝑛 3𝐴 =
1 − 3𝑡𝑎𝑛2 𝐴
𝑐𝑜𝑠(𝐴 + 𝐵) = 𝑐𝑜𝑠 𝐴 𝑐𝑜𝑠 𝐵 − 𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 𝐵
Sine and Cosine formulae
𝑎
𝑏
𝑐
=
=
𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 𝐵 𝑠𝑖𝑛 𝐶
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 𝑐𝑜𝑠 𝐴
𝑐𝑜𝑠(𝐴 − 𝐵) = 𝑐𝑜𝑠 𝐴 𝑐𝑜𝑠 𝐵 + 𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 𝐵
𝑡𝑎𝑛(𝐴 + 𝐵) =
𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵
1 − 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵
𝑡𝑎𝑛(𝐴 − 𝐵) =
𝑡𝑎𝑛 𝐴 − 𝑡𝑎𝑛 𝐵
1 + 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵
Product to sum formulae
Sum to product formulae
1
𝑠𝑖𝑛 𝐴 𝑐𝑜𝑠 𝐵 = [𝑠𝑖𝑛(𝐴 + 𝐵) + 𝑠𝑖𝑛(𝐴 − 𝐵)]
2
𝑠𝑖𝑛 𝐴 + 𝑠𝑖𝑛 𝐵 = 2 (𝑠𝑖𝑛
𝐴+𝐵
𝐴−𝐵
𝑐𝑜𝑠
)
2
2
1
𝑐𝑜𝑠 𝐴 𝑠𝑖𝑛 𝐵 = [𝑠𝑖𝑛(𝐴 + 𝐵) − 𝑠𝑖𝑛(𝐴 − 𝐵)]
2
𝑠𝑖𝑛 𝐴 − 𝑠𝑖𝑛 𝐵 = 2 (𝑐𝑜𝑠
𝐴+𝐵
𝐴−𝐵
𝑠𝑖𝑛
)
2
2
𝑐𝑜𝑠 𝐴 𝑐𝑜𝑠 𝐵 =
1
[𝑐𝑜𝑠(𝐴 + 𝐵) + 𝑐𝑜𝑠(𝐴 − 𝐵)]
2
1
𝑠𝑖𝑛 𝐴 𝑠𝑖𝑛 𝐵 = − [𝑐𝑜𝑠(𝐴 + 𝐵) − 𝑐𝑜𝑠(𝐴 − 𝐵)]
2
𝑐𝑜𝑠 𝐴 + 𝑐𝑜𝑠 𝐵 = 2 (𝑐𝑜𝑠
𝐴+𝐵
𝐴−𝐵
𝑐𝑜𝑠
)
2
2
𝑐𝑜𝑠 𝐴 − 𝑐𝑜𝑠 𝐵 = −2 (𝑠𝑖𝑛
𝐴+𝐵
𝐴−𝐵
𝑠𝑖𝑛
)
2
2
Differentiation and Integration formulae
Limit
𝑙𝑖𝑚
1
=0
𝑥→∞ 𝑥
𝑒𝑥 − 1
𝑙𝑖𝑚
=1
𝑥→0
𝑥
1
=0
𝑥→−∞ 𝑥
1 𝑥
𝑙𝑖𝑚 (1 + ) = 𝑒
𝑥→∞
𝑥
1 𝑥
𝑙𝑖𝑚 (1 + ) = 1
𝑥→0
𝑥
𝑙𝑖𝑚(1 + 𝑥)𝑥 = 𝑒
𝑙𝑖𝑚
𝑙𝑖𝑚
𝑥→0
𝑠𝑖𝑛 𝑥
𝑥
= 1 ; 𝑙𝑖𝑚
1
𝑥→0
𝑡𝑎𝑛 𝑥
𝑥→0
𝑥
=1
𝑎 𝑥
𝑙𝑖𝑚 (1 + ) = 𝑒 𝑎
𝑥→∞
𝑥
𝑙𝑖𝑚 𝑎 𝑥 = 0 where 0 < 𝑎 < 1
𝑥→∞
𝑙𝑖𝑚 𝑎 𝑥 = ∞ where 0 < 𝑎 < 1
𝑥→−∞
𝑙𝑖𝑚 𝑎 𝑥 = ∞ where 𝑎 > 1
𝑥→∞
𝑙𝑖𝑚 𝑎 𝑥 = 0 where 𝑎 > 1
𝑥→−∞
Differentiation formula
𝑑 𝑛
𝑥 = 𝑛𝑥 𝑛−1
𝑑𝑥
𝑑 𝑥
𝑎 = 𝑎 𝑥 𝑙𝑛 𝑎
𝑑𝑥
𝑑
𝑑𝑣
𝑑𝑢
𝑢𝑣 = 𝑢
+ 𝑣
𝑑𝑥
𝑑𝑥
𝑑𝑥
𝑑
𝑑𝑢 𝑑𝑣
(𝑢 + 𝑣) =
+
𝑑𝑥
𝑑𝑥 𝑑𝑥
𝑑 𝑥
𝑒 = 𝑒𝑥
𝑑𝑥
𝑑𝑢
𝑑𝑣
𝑣
−𝑢
𝑑 𝑢
( ) = 𝑑𝑥 2 𝑑𝑥
𝑑𝑥 𝑣
𝑣
𝑑
𝑑𝑢 𝑑𝑣
(𝑢 − 𝑣) =
−
𝑑𝑥
𝑑𝑥 𝑑𝑥
𝑑
1
𝑙𝑛 𝑥 =
𝑑𝑥
𝑥
𝑑
𝑑𝑥
(𝑥 𝑝 − 𝑥 + 𝑐)𝑛 = 𝑛(𝑥 𝑝 − 𝑥 + 𝑐)𝑛−1 (𝑝𝑥 𝑝−1 − 1) where p and n are any integers
Integration formulae
∫ 𝑥 𝑛 𝑑𝑥 =
𝑥 𝑛+1
+𝐶
𝑛+1
∫
1
𝑑𝑥 = 𝑙𝑛 𝑥 + 𝐶
𝑥
𝑎𝑥
+𝐶
𝑙𝑛 𝑎
∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶
∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢
∫ 𝑎 𝑥 𝑑𝑥 =
Trigonometry differentiation formula
Trigonometry integration formulae
𝑑
𝑠𝑖𝑛 𝑥 = 𝑐𝑜𝑠 𝑥
𝑑𝑥
∫ 𝑐𝑜𝑠 𝑥 𝑑𝑥 = 𝑠𝑖𝑛 𝑥 + 𝐶
∫ 𝑠𝑒𝑐 𝑥 𝑑𝑥 = 𝑙𝑛|𝑠𝑒𝑐 𝑢 + 𝑡𝑎𝑛 𝑢| + 𝐶
𝑑
𝑐𝑜𝑠 𝑥 = − 𝑠𝑖𝑛 𝑥
𝑑𝑥
∫ 𝑠𝑖𝑛 𝑥 𝑑𝑥 = − 𝑐𝑜𝑠 𝑥 + 𝐶
∫ 𝑐𝑠𝑐 𝑥 𝑑𝑥 = −𝑙𝑛|𝑐𝑠𝑐 𝑥 + 𝑐𝑜𝑡 𝑥| + 𝐶
𝑑
𝑡𝑎𝑛 𝑥 = 𝑠𝑒𝑐 2 𝑥
𝑑𝑥
∫ 𝑠𝑒𝑐 2 𝑥 𝑑𝑥 = 𝑡𝑎𝑛 𝑥 + 𝐶
∫ 𝑡𝑎𝑛 𝑥 𝑑𝑥 = 𝑙𝑛|𝑠𝑒𝑐 𝑥| + 𝐶
𝑑
𝑐𝑜𝑡 𝑥 = − 𝑐𝑠𝑐 2 𝑥
𝑑𝑥
∫ 𝑐𝑠𝑐 2 𝑥 𝑑𝑥 = −𝑐𝑜𝑡𝑥 + 𝐶
∫ 𝑐𝑜𝑡 𝑥 𝑑𝑥 = 𝑙𝑛|𝑠𝑖𝑛 𝑥| + 𝐶
𝑑
𝑠𝑒𝑐 𝑥 = 𝑡𝑎𝑛 𝑥 𝑠𝑒𝑐 𝑥
𝑑𝑥
∫ 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥 𝑑𝑥 = 𝑠𝑒𝑐 𝑥 + 𝐶
∫
𝑑
𝑐𝑠𝑐 𝑥 = − 𝑐𝑜𝑡 𝑥 𝑐𝑠𝑐 𝑥
𝑑𝑥
∫ 𝑐𝑜𝑡 𝑥 𝑐𝑠𝑐 𝑥 𝑑𝑥 = − 𝑐𝑠𝑐 𝑥 + 𝐶
∫
∫ 𝑙𝑛 𝑥 𝑑𝑥 = 𝑥 𝑙𝑛 𝑥 − 𝑥 + 𝐶
∫
𝑑𝑥
√𝑎2
−
𝑥2
𝑥
= 𝑠𝑖𝑛 −1 ( ) + 𝐶
𝑎
𝑑𝑥
𝑥
= 𝑡𝑎𝑛−1 ( ) + 𝐶
𝑎2 + 𝑥 2
𝑎
𝑑𝑥
𝑥√𝑥 2
−
𝑎2
𝑥
= 𝑠𝑒𝑐 −1 | | + 𝐶
𝑎