en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Popular Examples

sum from n=4 to infinity of (1/4)^n

n = 4 \sum \infty (\frac{1}{4})^{n}

integral of (2x+3)/((x-3)(x+3))

\int \frac{2 x + 3}{( x - 3 ) ( x + 3 )} d x

(\partial)/(\partial x)(tan(y)e^x)

\frac{\partial}{\partial x} (tan (y) e^{x})

limit as x approaches 1 of x^2-2x+1

x \to 1 lim (x^{2} - 2 x + 1)

sum from n=0 to infinity of 6^{n+1}

n = 0 \sum \infty 6^{n + 1}

Frequently Asked Questions (FAQ)

What is taylor |sin(x)| ?
The answer to taylor |sin(x)| is