For x = 0, let theta = sin⁻¹(x) + cos⁻¹(x) - tan⁻¹(x). Which range does theta lie in?

Question

For x >= 0, let theta = sin⁻¹(x) + cos⁻¹(x) - tan⁻¹(x). Which range does theta lie in?

Accepted Answer

pi/4 <= theta <= pi/2. Since both sin⁻¹ and cos⁻¹ require x in [-1,1] and here x>=0, x is in [0,1]. Identity: sin⁻¹(x)+cos⁻¹(x) = pi/2. So theta = pi/2 - tan⁻¹(x). For x in [0,1], tan⁻¹(x) ranges in [0, pi/4]. Hence theta ranges from pi/2 - pi/4 = pi/4 (at x=1) up to pi/2 - 0 = pi/2 (at x=0). Thus pi/4 <= theta <= pi/2.

For x >= 0, let theta = sin⁻¹(x) + cos⁻¹(x) - tan⁻¹(x). Which range does theta lie in?

Solution

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