Inverse Trigonometric Functions

INVERSE TRIGONOMETRIC (CIRCULAR) FUNCTIONS:

If sinθ = x , then q may be any angle whose sine is x, and we write θ = sin-1x . It means that θ is an angle which can be determined from its sine. Thus tan-1 (1/√3) is an angle whose tangent is 1/√3 ,
.e. 1/√3 = tan-1 = nπ + π/6  , where π/6 is the least positive value of  θ .

The functions sin-1x, cos-1x, tan-1x, cot-1x, cosec-1x and sec-1x are called inverse circular or inverse trigonometric functions.

Each of the inverse circular function is multivalued (infact they are relations). To make each inverse circular function single valued we define principal value as follows. If x is positive, the principal values of all the inverse circular functions lie between 0 and π/2 . If x is negative, the principal values of sin-1x , cosec-1x and tan-1x lie between -π/2 and 0 . Those of cos-1x, sec-1x and cot-1x lie between π/2 and π . From now onwards we take only principal values.

sinθ = x ⇒ θ = sin-1x

Where θ ∈ [-π/2 , π/2] and x ∈ [-1, 1].

cosecθ = x ⇒ θ = cosec-1x

Where θ ∈ [ -π/2 , 0 ) ∪ (0 , π/2  ] and x ∈ (-∞ , -1] ∪ [1 , ∞)

tanθ = x ⇒ θ = tan-1x

where  θ ∈ (-π/2 , π/2 )and x ∈ (-∞, ∞)

cos θ = x ⇒ θ = cos-1x

where θ ∈ [0, π] and x ∈ [-1, 1]

secθ = x ⇒ θ = sec-1x

where θ ∈ [0 , π/2) ∪(π/2 , π] and x ∈ (-∞, -1] ∪ [1 , ∞)

cotθ = x ⇒ θ = cot-1x

where θ ∈ (0, π) and x ∈ (-∞, ∞)