J's answer to Rella's Junior College 1 H2 Maths Singapore question.
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From the earlier part you would obtain y ≈ 1 - x²/4
So (cos x)¹/² ≈ 1 - x²/4
We see a 1/⁴√2 on the left hand side so this gives us a clue :
To use a value of x such that cos x = 1/√2 ,which in turn implies that (cos x)¹/² = (1/√2)¹/² = 1/(√2)¹/² = 1/(2¹/²)¹/² = 1 / 2¹/⁴ = 1/⁴√2
The approximation is closer as x gets closer to 0 . Since cos π/4 = 1/√2, use the substitution x = π/4 (≈ 0.785)
So (cos π/4)¹/² ≈ 1 - (π/4)²/4
1/⁴√2 ≈ 1 - π²/64
1/⁴√2 = 1 + (-1/64) π²
So k = -1/64
So (cos x)¹/² ≈ 1 - x²/4
We see a 1/⁴√2 on the left hand side so this gives us a clue :
To use a value of x such that cos x = 1/√2 ,which in turn implies that (cos x)¹/² = (1/√2)¹/² = 1/(√2)¹/² = 1/(2¹/²)¹/² = 1 / 2¹/⁴ = 1/⁴√2
The approximation is closer as x gets closer to 0 . Since cos π/4 = 1/√2, use the substitution x = π/4 (≈ 0.785)
So (cos π/4)¹/² ≈ 1 - (π/4)²/4
1/⁴√2 ≈ 1 - π²/64
1/⁴√2 = 1 + (-1/64) π²
So k = -1/64
Date Posted:
3 years ago
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