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Home Trigonometric Functions Inverse Trigonometric Functions Theorem 2: | |
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Theorem 2:
THEOREM 2 (i) (where - 1 < x < 1). (where - 1 < x < 1). (ii)
(iii) (where |x| > 1). (where |x| > 1). PROOF We prove the first part of (i) and (iii). Since the derivatives exist we may use implicit differentiation. (i) Let y = arcsin x. Then x = sin y, -π/2 ≤ y ≤ π/2, dx = cos y dy. From sin2 y + cos2 y = 1 we get Since - π/2 ≤ y ≤ π/2, cos y ≥ 0. Then Substituting, (iii) Let y = arcsec x. Then x = sec y, 0 ≤ y ≤ π, dx = sec y tan y dy. From tan2 y + 1 = sec2 y we get tan y = = Since 0 ≤ y ≤ π, tan y and sec have the same sign. Therefore sec y tan y ≥ 0 and When we turn these formulas for derivatives around we get some surprising new integration formulas.
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