Find the Value of Sin - Edubirdie

University:
University of Cambridge
Course:
0580 IGCSE | Mathematics
Academic year:

2023
Views:

146

Pages:

3
Author:

Shahbaz A.

Calculus shahbaz ahmed August 2024 Relevant skills If x = tan θ , find the value of sin θ Solution x = tan θ x2 = tan2 θ 1 + x2 = 1 + tan2 θ 1 + x2 = sec2 θ 1 + x2 = cos2 θ = 1 cos2 θ 1 1+x2 1 − sin2 θ = 1− 1 1+x2 1 1+x2 = sin2 θ 1+x2 −1 1+x2 = sin2 θ q √ x2 sin2 θ = 1+x2 √ x 1+x2 = sin θ 1 ............................................. Find Z dx 3 (1 + x2 ) 2 Solution Z dx 3 (1 + x2 ) 2 (1) Putting x = tan θ x2 = tan2 θ 1 + x2 = 1 + tan2 θ 1 + x2 = sec2 θ 3 3 (1 + x2 ) 2 = (sec2 θ) 2 3 (1 + x2 ) 2 = sec3 θ Similarly x = tan θ dx dθ = dx dθ = sec2 θ d dθ (tan θ) 2 (2) dx = sec2 θ dθ Putting equation (2) and equation (3) in equation (1) sec2 θ dθ 3 = sec3 θ (1 + x2 ) 2 Z Z dθ dx 3 = sec θ (1 + x2 ) 2 Z Z dx cos θ dθ 3 = (1 + x2 ) 2 Z dx 3 = sin θ + C (1 + x2 ) 2 Z Putting sin θ = Z dx √ x 1+x2 Z dx (1 + x2 ) 3 2 =√ 3 x +C 1 + x2 (3)

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