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Popular Calculus >

taylor |sin(x)|

Solution

taylor

∣ sin (x) ∣

Solution

sin (1) + cos (1) (x - 1) + \frac{cos ( 2 ) - cos ^{2} ( 1 )}{2 sin ( 1 )} (x - 1)^{2} - \frac{sin ( 2 ) sin ( 1 ) + cos ( 1 ) cos ( 2 ) - cos ^{3} ( 1 )}{6 sin ^{2} ( 1 )} (x - 1)^{3} - \frac{3 sin ^{2} ( 1 ) cos ^{2} ( 1 ) - sin ^{2} ( 1 ) - 2 cos ^{2} ( 1 ) cos ( 2 ) + 2 cos ^{4} ( 1 ) - 2 cos ( 1 ) sin ( 2 ) sin ( 1 )}{24 sin ^{3} ( 1 )} (x - 1)^{4} + \dots

Solution steps

Apply the Taylor Formula

= sin (1) + \frac{\frac{d}{d x} ( ∣ sin ( x ) ∣ ) ( 1 )}{1 !} (x - 1) + \frac{\frac{d ^{2}}{d x ^{2}} ( ∣ sin ( x ) ∣ ) ( 1 )}{2 !} (x - 1)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ∣ sin ( x ) ∣ ) ( 1 )}{3 !} (x - 1)^{3} + \dots

Evaluate Derivatives

= sin (1) + \frac{cos ( 1 )}{1 !} (x - 1) + \frac{\frac{c o s ( 2 ) - c o s ^{2} ( 1 )}{s i n ( 1 )}}{2 !} (x - 1)^{2} + \frac{- \frac{s i n ( 2 ) s i n ( 1 ) + c o s ( 1 ) c o s ( 2 ) - c o s ^{3} ( 1 )}{s i n ^{2} ( 1 )}}{3 !} (x - 1)^{3} + \frac{- \frac{3 s i n ^{2} ( 1 ) c o s ^{2} ( 1 ) - s i n ^{2} ( 1 ) - 2 c o s ^{2} ( 1 ) c o s ( 2 ) + 2 c o s ^{4} ( 1 ) - 2 c o s ( 1 ) s i n ( 2 ) s i n ( 1 )}{s i n ^{3} ( 1 )}}{4 !} (x - 1)^{4} + \dots

Refine

= sin (1) + cos (1) (x - 1) + \frac{cos ( 2 ) - cos ^{2} ( 1 )}{2 sin ( 1 )} (x - 1)^{2} - \frac{sin ( 2 ) sin ( 1 ) + cos ( 1 ) cos ( 2 ) - cos ^{3} ( 1 )}{6 sin ^{2} ( 1 )} (x - 1)^{3} - \frac{3 sin ^{2} ( 1 ) cos ^{2} ( 1 ) - sin ^{2} ( 1 ) - 2 cos ^{2} ( 1 ) cos ( 2 ) + 2 cos ^{4} ( 1 ) - 2 cos ( 1 ) sin ( 2 ) sin ( 1 )}{24 sin ^{3} ( 1 )} (x - 1)^{4} + \dots

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Frequently Asked Questions (FAQ)

What is taylor |sin(x)| ?
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