If theta =[2pie÷7],then find the value of sec theta +sec 2theta +sec 4theta

Question

Answer

Solution:

Given, theta = 2π/7

Using the trigonometric identity:

sec²θ = 1 + tan²θ

tanθ = sinθ/cosθ = (2 sin(π/7))/cos(2π/7) (using double angle formula)

Squaring and adding numerator and denominator, we get:

tan²θ = (4 sin²(π/7))/(1 - 2 sin²(π/7))

Therefore, sec²θ = 1 + (4 sin²(π/7))/(1 - 2 sin²(π/7))

secθ = ±√(1 + (4 sin²(π/7))/(1 - 2 sin²(π/7)))

Since theta lies in the first quadrant, secθ is positive:

secθ = √(1 + (4 sin²(π/7))/(1 - 2 sin²(π/7))) ≈ 1.802

Using the double angle formula:

sec(2θ) = 1/cos(2θ) = 1/(2 cos²θ - 1)

Substituting the value of secθ we obtained earlier:

sec(2θ) = 1/(2(√(1 + (4 sin²(π/7))/(1 - 2 sin²(π/7))))² - 1) ≈ 3.732

Using the double angle formula:

sec(4θ) = 1/cos(4θ) = 1/(8 cos⁴θ - 8 cos²θ + 1)

Substituting the value of secθ we obtained earlier:

sec(4θ) = 1/(8(√(1 + (4 sin²(π/7))/(1 - 2 sin²(π/7))))⁴ - 8(√(1 + (4 sin²(π/7))/(1 - 2 sin²(π/7))))² + 1) ≈ 11.591

Answer

Theta=[2π/7]=[6.28/7]=0
=> sec0 + sec(2*0) + sec(4*0) = 1+1+1=3