Section 10.6: The Inverse Trigonometric Functions - Ostts.org

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854 Foundations of Trigonometry172. cos (2 arctan (x)) = 1 − x2for all x1 + x2 173. sin(arccos(2x)) = √ 1 − 4x 2 for − 1 2 ≤ x ≤ 1 2( ( x))√25 − x 2174. sin arccos = for −5 ≤ x ≤ 55 5( ( x))√4 − x 2175. cos arcsin = for −2 ≤ x ≤ 22 2176. cos (arctan (3x)) =1√1 + 9x 2for all x177. sin(2 arcsin(7x)) = 14x √ 1 − 49x 2 for − 1 7 ≤ x ≤ 1 7( (x √ ))3178. sin 2 arcsin = 2x√ 3 − x 2for − √ 3 ≤ x ≤ √ 333179. cos(2 arcsin(4x)) = 1 − 32x 2 for − 1 4 ≤ x ≤ 1 4180. sec(arctan(2x)) tan(arctan(2x)) = 2x √ 1 + 4x 2 for all x181. sin (arcsin(x) + arccos(x)) = 1 for −1 ≤ x ≤ 1√1 − x182. cos (arcsin(x) + arctan(x)) =2 − x 2√ for −1 ≤ x ≤ 11 + x 2(183. 10 tan (2 arcsin(x)) = 2x√ √ ) ( √ √ ) (√ )1 − x 22 2 2 21 − 2x 2 for x in −1, − ∪ −2 2 , ∪2 2 , 1( )1184. sin2 arctan(x) =⎧⎪⎨⎪⎩−√ √x 2 + 1 − 12 √ x 2 + 1√ √x 2 + 1 − 12 √ x 2 + 1for x ≥ 0for x < 0185. If sin(θ) = x 2 for −π 2 < θ < π 2 , then θ + sin(2θ) = arcsin ( x2186. If tan(θ) = x 7 for −π 2 < θ < π 2 , then 1 2 θ − 1 2 sin(2θ) = 1 2 arctan ( x7)+ x√ 4 − x 22)−7xx 2 + 4910 The equivalence for x = ±1 can be verified independently of the derivation of the formula, but Calculus is requiredto fully understand what is happening at those x values. You’ll see what we mean when you work through the detailsof the identity for tan(2t). For now, we exclude x = ±1 from our answer.
10.6 The Inverse Trigonometric Functions 855187. If sec(θ) = x 4 for 0 < θ < π 2 , then 4 tan(θ) − 4θ = √ ( x)x 2 − 16 − 4arcsec4( ( 7 7188. x = arcsin + 2πk or x = π − arcsin + 2πk, in [0, 2π), x ≈ 0.6898, 2.451811)11)(189. x = arccos − 2 )(+ 2πk or x = − arccos − 2 )+ 2πk, in [0, 2π), x ≈ 1.7949, 4.488399190. x = π + arcsin(0.569) + 2πk or x = 2π − arcsin(0.569) + 2πk, in [0, 2π), x ≈ 3.7469, 5.6779191. x = arccos(0.117) + 2πk or x = 2π − arccos(0.117) + 2πk, in [0, 2π), x ≈ 1.4535, 4.8297192. x = arcsin(0.008) + 2πk or x = π − arcsin(0.008) + 2πk, in [0, 2π), x ≈ 0.0080, 3.1336( )( )359359193. x = arccos + 2πk or x = 2π − arccos + 2πk, in [0, 2π), x ≈ 0.0746, 6.2086360360194. x = arctan(117) + πk, in [0, 2π), x ≈ 1.56225, 4.70384(195. x = arctan − 1 )+ πk, in [0, 2π), x ≈ 3.0585, 6.200012196. x = arccos197. x = π + arcsin( 23)+ 2πk or x = 2π − arccos( 23)+ 2πk, in [0, 2π), x ≈ 0.8411, 5.4422( )( )1717+ 2πk or x = 2π − arcsin + 2πk, in [0, 2π), x ≈ 3.3316, 6.09329090198. x = arctan ( − √ 10 ) + πk, in [0, 2π), x ≈ 1.8771, 5.0187( ( 3 3199. x = arcsin + 2πk or x = π − arcsin + 2πk, in [0, 2π), x ≈ 0.3844, 2.75728)8)(200. x = arccos − 7 )(+ 2πk or x = − arccos − 7 )+ 2πk, in [0, 2π), x ≈ 2.0236, 4.25961616201. x = arctan(0.03) + πk, in [0, 2π), x ≈ 0.0300, 3.1716202. x = arcsin(0.3502) + 2πk or x = π − arcsin(0.3502) + 2πk, in [0, 2π), x ≈ 0.3578, 2.784203. x = π + arcsin(0.721) + 2πk or x = 2π − arcsin(0.721) + 2πk, in [0, 2π), x ≈ 3.9468, 5.4780204. x = arccos(0.9824) + 2πk or x = 2π − arccos(0.9824) + 2πk, in [0, 2π), x ≈ 0.1879, 6.0953205. x = arccos(−0.5637) + 2πk or x = − arccos(−0.5637) + 2πk, in [0, 2π), x ≈ 2.1697, 4.1135206. x = arctan(117) + πk, in [0, 2π), x ≈ 1.5622, 4.7038207. x = arctan(−0.6109) + πk, in [0, 2π), x ≈ 2.5932, 5.7348
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Section 10.6: The Inverse Trigonometric Functions - Ostts.org

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