SOLUTION: Use the Maclaurin expansion of sin x to find the value of sin 45 correct to four decimal places

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Question 893601: Use the Maclaurin expansion of sin x to find the value of sin 45 correct to four decimal places
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The Maclaurin series expansion is,
f%28x%29=sum%28f%5E%28k%29%280%29%2A%28%28x-0%29%5Ek%2Fk%21%29%2Ck=0%2Cinfinity%29
where f%5Ek is the k-th derivative of the function.
In this case,
f%28x%29=sin%28x%29
f%5E%281%29%28x%29=cos%28x%29
f%5E%282%29%28x%29=-sin%28x%29
f%5E%283%29%28x%29=-cos%28x%29
f%5E%284%29%28x%29=sin%28x%29
f%5E%285%29%28x%29=cos%28x%29
f%5E%286%29%28x%29=-sin%28x%29
Then take the value of each at x=0
f%280%29=sin%280%29=0
f%5E%281%29%280%29=cos%280%29=1
f%5E%282%29%280%29=-sin%280%29=0
f%5E%283%29%280%29=-cos%280%29=-1
f%5E%284%29%280%29=sin%280%29=0
f%5E%285%29%280%29=cos%280%29=1
f%5E%286%29%280%29=-sin%280%29=0
Putting it all together,

You begin to see the pattern,
sin%28x%29=x-%28x%29%5E3%2F3%21%2B%28x%29%5E5%2F5%21-%28x%29%5E7%2F7%21
So find the value of 45 degrees in radians,
45%28%282%2Api%29%2F360%29=0.785398
And substitute,

sin%280.785398%29=0.707106
I think that's pretty good compared to sqrt%282%29%2F2=0.707107
sin%280.785398%29=0.7071