Part IV. Inverse Trigonometric Function
Properties of sin-1(x), cos-1(x), tan-1(x): inverses of sin(x), cos(x), tan(x) with restricted domains
● sin[arcsin(x)]=x; x ∈[-π/2, π/2]
arcsin[sin(x)]=x; x ∈[-π/2, π/2]
● cos[arccos(x)]=x; x ∈[0,π]
arccos[cos(x)]=x; x ∈[0,π]
● tan[arctan(x)]=x; x ∈(-π/2, π/2)
arctan[tan(x)]=x; x ∈(-π/2, π/2)
Solutions of Trigonometric Equations
1. sin(𝜃)=y
𝜃=sin-1(y)+2kπ, k ∈Z
𝜃=π-sin-1(y)+2kπ, k ∈Z
2. cos(𝜃)=x
𝜃=cos-1(x)+2kπ, k ∈Z
𝜃=-cos-1(x)+2kπ, k ∈Z
3. tan(𝜃)=𝛼
𝜃=tan-1(𝛼)+kπ, k ∈Z
Identities of sin-1(x), cos-1(x), tan-1(x)
● arcsin(-x)=arcsin(x); for x in domain to enable arcsin(x) to exist
● arccos(-x)=π-arcsin(x)
● arctan(-x)=-arctan(x); for x in domain to enable arctan(x) to exist
Graphs of sin-1(x), cos-1(x), tan-1(x)
i. y=arcsin(x)
, ii. y=arccos(x)
iii. y=arctan(x)
Properties of sin-1(x), cos-1(x), tan-1(x): inverses of sin(x), cos(x), tan(x) with restricted domains
● sin[arcsin(x)]=x; x ∈[-π/2, π/2]
arcsin[sin(x)]=x; x ∈[-π/2, π/2]
● cos[arccos(x)]=x; x ∈[0,π]
arccos[cos(x)]=x; x ∈[0,π]
● tan[arctan(x)]=x; x ∈(-π/2, π/2)
arctan[tan(x)]=x; x ∈(-π/2, π/2)
Solutions of Trigonometric Equations
1. sin(𝜃)=y
𝜃=sin-1(y)+2kπ, k ∈Z
𝜃=π-sin-1(y)+2kπ, k ∈Z
2. cos(𝜃)=x
𝜃=cos-1(x)+2kπ, k ∈Z
𝜃=-cos-1(x)+2kπ, k ∈Z
3. tan(𝜃)=𝛼
𝜃=tan-1(𝛼)+kπ, k ∈Z
Identities of sin-1(x), cos-1(x), tan-1(x)
● arcsin(-x)=arcsin(x); for x in domain to enable arcsin(x) to exist
● arccos(-x)=π-arcsin(x)
● arctan(-x)=-arctan(x); for x in domain to enable arctan(x) to exist
Graphs of sin-1(x), cos-1(x), tan-1(x)
i. y=arcsin(x)
, ii. y=arccos(x)
iii. y=arctan(x)
