# Principal solution of an equation

## Principal solution of an equation (or Principal value of an angle)

i. The sine function from [– , ] → [– 1, 1] is bijective.
There exists a unique value 'α' in [– , ] satisfying sin α = k for k ∈ [– 1, 1].
This 'α' is called principal value or principal solution of 'θ' satisfying the equation sin θ = k.

ii. The cosine function from [0, π] → [– 1, 1] is bijective.
There exists a unique value 'α' in [0, π] satisfying cos α = k for k ∈ [– 1, 1].
This 'α' is called the principal solution of 'θ' satisfying the equation cos θ = k.

iii. The tangent function from (– , ) → (– ∞, ∞) is bijective.
There exists a unique value 'α' in (– , ) satisfying tan α = k for k ∈ (– ∞, ∞).
This 'α' is called the principal solution of 'θ' satisfying the equation tan θ = k.

iv. The principal value of θ satisfying the equation
a) cot θ = k, k ∈ (– ∞, ∞), lies in [– , ] – {0}
b) sec θ = k, k ∈ ( – ∞, – 1] ∪ [1, ∞) lies in [0, π] – { }
c) cosec θ = k, k ∈ ( – ∞, – 1] ∪ [1, ∞) lies in [– , ] – {0}.