If theta =[2pie÷7],then find the value of sec theta +sec 2theta +sec 4...
Solution:
- Finding the value of sec(theta):
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
- Finding the value of sec(2θ):
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
- Finding the value of sec(4θ):
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