Question 7
Prove that if 1/2 ≤ x ≤ 1 then cos−1 x + cos−1 [𝑥/2+√(3 − 3𝑥^2 )/2] = 𝜋/3
Solving LHS
cos−1 x + cos−1 [𝑥/2+√(3 − 3𝑥^2 )/2]
= cos−1 x + cos−1 [𝑥/2+√(3(1 − 𝑥^2))/2]
Let x = cos 𝜃
i.e. cos−1 x = 𝜃
Theory
Since it is √(1−𝑥^2 )
we can use x = sin 𝜃 or x = cos 𝜃
Since it is given cos-1 x,
we use x = cos 𝜃
= 𝜃 + cos−1 [cos𝜃/2+√(3(1−cos^2〖𝜃)〗 )/2]
Using sin θ = 1 – cos2 θ
= 𝜃 + cos−1 [cos𝜃/2+√(3 sin^2𝜃 )/2]
= 𝜃 + cos−1 [cos𝜃/2+〖√3 sin〗𝜃/2]
= 𝜃 + cos−1 [1/2 ×cos𝜃+√3/2 ×sin𝜃 ]
Theory
We convert 1/2 ×cos𝜃+√3/2 ×sin𝜃
into cos (x – y)
Since cos (x – y) = cos x cos y + sin x sin y
We use x = 60° and y = 𝜃
= 𝜃 + cos−1 [cos〖60°〗 ×cos𝜃+sin〖60°〗×sin𝜃 ]
= 𝜃 + cos−1 [cos〖𝜋/3〗 ×cos𝜃+sin〖𝜋/3〗×sin𝜃 ]
Using cos (x – y) = cos x cos y + sin x sin y
= 𝜃 + cos−1 [cos(𝜋/3−𝜃) ]
= 𝜃 + cos−1 [𝑐𝑜𝑠(𝜋/3−𝜃)]
= 𝜃 + 𝜋/3 − 𝜃
= 𝜋/3
= RHS
Hence proved

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!