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Sine, Cosine, Tangent, Exponential, and other functions come up all the time in engineering and science. When these functions appear, I find myself searching for a calculator. There is a way, however, to make quick estimates without a calculator. Try using the series expansions to get a value.
Step I - Memorize these 3 expansions
Keep in mind that for the Sine or Cosine above, if your value is in degrees you'll have to convert it to radians with the relation:
(π/180)=1° so 90°=(90π/180)=(π/2)
Step 2- Plug in your value
Plug the appropriate value for x and calculate as far as you want:
e1 = e = 1 + 1 + 1/2 + 1/6 + 1/24 + ...
Step 3 - Choose your accuracy
Each additional term you include will improve your accuracy, except for when x=0 and then only the first term is needed. For example, taking the exponential function out to 4 terms gives:
e1 = 1 + 1 + 1/2 + 1/6 ...~ 2.6666...
Adding an additional term gives:
e1 = e = 1 + 1 + 1/2 + 1/6 + 1/24 + ... ~ 1 + 1 + .5 + .1666 + .0416 ~ 2.708
The actual value is 2.71
So you can see that as you add terms the accuracy improves.
Here are some examples:
Sin 45° = Sin(π/4) = π/4 - (1/6) (π/4)3 + (1/120) (π/4)5 + ....
~ 3/4- (1/6)(27/64) ~ .75 - .07 ~ .68 (Actual Value = .707...)
Note that the smaller your x, the fewer number of terms you'll have to include to get an accurate answer. Take the extreme example:
Sin 0 = (0) - (1/6)(0)3 +... = 0
Notice that only the first term is needed to define it exactly. This also works for Cosine and the Exponential:
Cos 0 = 1 - 1/2(0)2 + ... = 1 - 0 = 1
e0 = 1 + (0) + ... = 1
Some less exact examples:
e.01=1 + (.01) + .5(.0001) + ... ~ 1.01005
Where 1.010050167.... is the actual answer, not bad. For the larger x below:
e100= 1 + (100) + 1/2 (100)2 + 1/6 (100)3 + ....... ~ 5x105
Where 26.88117142..x 1042 is the actual answer, yikes! Clearly in this case the trick isn't useful.
When solving equations, this feature for small x can be used to calculate "small angle" effects. For very small angles:
Sin x = x
Cos x = 1
Series Expansions also can make it easier to understand certain concepts. Take the derivative of the exponent for example:
(d/dx) ex = ex
in series form it's easy to see this is true:
(d/dx)ex=(d/dx) [1 + x + (1/2)x2 + (1/6)x3 +....]
= (d/dx)1 + (d/dx)x + (d/dx)(1/2)x2 + (d/dx)(1/6)x3
= 0 + 1 + x + (1/2)x2 + ... = ex
So if you run into Sines, Cosines, and Exponents frequently, you might want to learn these handy series expansions above. Till next time.
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