expand tan(x+pi/4) as far as the term X power 4 and evaluate tan 46.5 degree up to four significant digits
Answers
Answer: To evaluate: tan(x+π4)
We know that the radian value of π=1800
So, π4=18004=450
⇒ tanπ4 =tan450=1
We know that tan(A+B)=tanA+tanB1-tanAtanB
⇒ tan(x+π4)=tanx+tanπ41-tanxtanπ4
=tanx+tan4501-tanxtan450
=tanx+11-tanx×1
=1+tanx1-tanx
Thus, the required value of tan(x+π4)=1+tanx1-tanx.
Step-by-step explanation:
STEP:1 Firstly, set f(x) = tan(x). We can then differentiate this 3 times in order to find f'(x), f''(x) and f'''(x). This will require use of the chain rule and the product rule. We can find:f'(x) = sec2xf''(x) = 2 * sec(x) * sec(x)tan(x) = 2sec2(x)tan(x)f'''(x) = 2 * 2sec2(x)tan(x)tan(x) + 2sec2(x)*sec2(x) = 4sec2(x)tan2(x) + 2sec4(x)We then substitute x with π/4, and find the values of f at these values, and then the coefficient an:f(π/4) = 1 a0 = 1/0! = 1 f'(π/4) = 2 a1 = 2/1! = 2f''(π/4) = 4 a2 = 4/2! = 2f'''(π/4) = 16 a3 = 16/3! = 8/3From this we can conclude tan(x) = 1 + 2(x-π/4) + 2(x-π/4)2 + (8/3)(x-π/4)^3.
STEP:2 or example if you wanted tan 11⁰ = tan 0.2 rad then
the term in x^7 comes to 0.00000069 whereas in the case we are dealing with it comes to 0.0125…..
So whoever asked you to work out the value of tan 46.50 to 4 dp by using the Taylor series was being a bit unreasonable.
I suppose you could use the fact that tan (3Θ) = (3tan Θ - tan³ Θ)/(1 – 3tan² Θ) to work out tan Θ = tan 15.5 = tan 0.2705 rad which does converge rapidly to 0.277324 and which, using the identity above, gives tan (3Θ) = 1.0538 to 4 dp but that is a lot of work for little return.
Perhaps someone cleverer than me can suggest a better method.
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