Find the values of the following:tan-(1)+ cosi12

1.	Find the values of the following:tan-(1)+ cosi12
Answer» We know that the range of principal values of tan^-1, cos^-1 and sin^-1 are (−π/2,π/2) , (0,π) and (−π/2,π/2) respectively, Let tan^-1(1) = θ1. Then, tanθ1 = 1 = tanπ/4 ⇒θ1 = π/4 ∈[0, π]. Let cos^−1(−1/2)=θ2. Then,cos θ2=−1/2 = −cos π/3=cos(π− π/3)=cos 2π/3So θ2 =2π/3∈[0 , π]. Let sin^−1(−1/2)=θ3. Then, sin θ3=−1/2 = −sin π/6 =sin (−π/6)So θ3 =−π/6∈[−π2 , π2]. So, tan^−1(1)+cos^−1(−12)+sin^−1(−12)= π/4+ 2π/3−π/6=3π/4

Answer»

We know that the range of principal values of tan^-1, cos^-1 and sin^-1 are (−π/2,π/2) , (0,π) and (−π/2,π/2) respectively,

Let tan^-1(1) = θ1. Then, tanθ1 = 1 = tanπ/4 ⇒θ1 = π/4 ∈[0, π].

Let cos^−1(−1/2)=θ2. Then,cos θ2=−1/2 = −cos π/3=cos(π− π/3)=cos 2π/3So θ2 =2π/3∈[0 , π].

Let sin^−1(−1/2)=θ3. Then, sin θ3=−1/2 = −sin π/6 =sin (−π/6)So θ3 =−π/6∈[−π2 , π2].

So, tan^−1(1)+cos^−1(−12)+sin^−1(−12)= π/4+ 2π/3−π/6=3π/4

Discussion