MTH 251 – Differential Calculus Chapter 3 – DifferentiationTrigonometry Review - InversesSlide 3Slide 4Compositions of Trig and Inverse-Trig FunctionsSlide 6Derivative of y = sin -1xDerivatives of the Inverse Trigonometric FunctionsMTH 251 – Differential CalculusChapter 3 – DifferentiationSection 3.8Inverse Trigonometric FunctionsTrigonometry Review - Inverses•The sine function … y = sin xNOTE: This is NOT 1-1.•A sine function w/ limited domain: –/2 ≤ x ≤ /2The blue curve is 1-1 and therefore has an inverse.The inverse is called: y = arcsin x = sin -1x Domain: –1 ≤ x ≤ 1Range: –/2 ≤ y ≤ /2Trigonometry Review - Inversesy = arcsin x = sin -1xDomain: –1 ≤ x ≤ 1Range: –/2 ≤ y ≤ /2y = arccos x = cos -1xDomain: –1 ≤ x ≤ 1Range: 0 ≤ y ≤ y = arctan x = tan -1xDomain: – ≤ x ≤ Range: –/2 ≤ y ≤ /2Trigonometry Review - Inversesy = arccsc x = csc -1x = sin -1(1/x)Domain: x ≤ –1 or x ≥ 1Range: –/2 ≤ y ≤ /2 , y 0y = arcsec x = sec -1x = cos -1(1/x)Domain: x ≤ –1 or x ≥ 1Range: 0 ≤ y ≤ , y /2y = arccot x = cot -1x = tan -1(1/x)Domain: – ≤ x ≤ Range: 0 ≤ y ≤ Compositions of Trig and Inverse-Trig Functions•Example …•In general …1cot(cot )x-x=1( )trig trig x-x=NOTE: There may be restrictions on x due to the inverse function in use.Compositions of Trig and Inverse-Trig Functions•Example …1cot(sin )x-21 xx-=1sin x-21 x-1xNOTE: There may be restrictions on x due to the inverse function in use.Derivative of y = sin -1x1sindxdx-� �� �[ ]1sin1sinxdxdx-=1sin1cosxx-=11cos(sin )x-=211 x=-Derivatives of the Inverse Trigonometric Functions1 121sin cos1d dx xdx dxx- -� � � �= =-� � � �-1 121tan cot1d dx xdx x dx- -� � � �= =-� � � �+1 121sec csc1d dx xdx dxx x- -� � � �= =-� � � �-Yes, you NEED to memorize
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