SOME USEFUL
FORMULAS
For use with 𝑥 = sin 𝜃 ∶
sin(sin−1 𝑥) = 𝑥,
sec(sin−1 𝑥) =
tan(sin−1 𝑥) =
−1≤𝑥 ≤1
1
√1 − 𝑥 2
𝑥
√1 −
𝑥2
,
−1<𝑥 <1
,
−1<𝑥 <1
For use with 𝑥 = tan 𝜃 ∶
sin(tan−1 𝑥) =
𝑥
√1 + 𝑥 2
,
sec(tan−1 𝑥) = √1 + 𝑥 2 ,
tan(tan−1 𝑥) = 𝑥,
For use with 𝑥 = sec 𝜃 ∶
sin(sec −1 𝑥) =
√𝑥 2 − 1
√𝑥 2 − 1
𝑥2 − 1
= 𝑠𝑔𝑛(𝑥)
=√ 2 ,
|𝑥|
𝑥
𝑥
sec(sec −1 𝑥) = 𝑥,
tan(sec
−1
𝑥) =
𝑠𝑔𝑛(𝑥)√𝑥 2
𝑥 ≤ −1 𝑜𝑟 1 ≤ 𝑥
𝑥 ≤ −1 𝑜𝑟 1 ≤ 𝑥
𝑥2 − 1
√
−1=𝑥
, 𝑥 ≤ −1 𝑜𝑟 1 ≤ 𝑥
𝑥2