Best way to write formulae or equations ?

[rollcast]

When I'm working out a maths or physics problem I always find myself questioning whether I should work out the equation as I go along or should I re arrange it and do it in one step with my calculator?

For example if I was to use the cosine rule for triangles,

a²=b²+c²-2bccosA

to solve a triangle for angle A with sides a, b and c as 20, 30 and 40 respectively.

Is this better written out as,

a²=b²+c²-2bccosA
20²=30²+40²-2*30*40*cosA
400=900+1600-2400cosA
-2100=-2400cosA
7/8=cosA
A=cos^-1(7/8)
A=29.0°(1D.P.)

or

a²=b²+c²-2bccosA
A=cos^-1\left(\frac{b^{2}+c^{2}-a^{2}}{2bc}\right)
A=cos^-1\left(\frac{30^{2}+40^{2}-20^{2}}{2*30*40}\right)
A=cos^-1(7/8)
A=29.0(1D.P.)

Thanks
AL

 

[whizz]

I'd go with the second approach because arranging everything into an expression as a whole and simplifying it, then directly getting the result would be rather easy.

 

[CompuChip]

I prefer the latter, for several reasons:
1) it let's you use your algebraic skills
2) it gives fewer rounding errors, since you don't have intermediary results
3) if the values have units - e.g. in physics or even here (a, b, c are lengths) - you can easily check that the final units are correct before plugging in the numbers, more easily catching any errors.

 

[Mentallic]

For me, it usually depends on the calculator I'm using. If I'm forced to use one of the older calculators that can only use brackets and don't have the fancy displays to express a fraction as it looks on paper, then the brackets could quickly get out of hand and it leaves you more prone to making errors in that way, so I'd use your first approach in that case.

Also I've noticed that on my calculator with its fancy display, it tends to lag when a lot has been input into one display, so if it saves time, I'd make some quick and easy mental calculations to simplify things before entering it in, for example,

30²+40²=50²
2*30*40=2*3*4*100=24*100=2400

 

[rollcast]

Thanks guys,

In school when we where introduced to algebra we were taught to use the first method of solving various parts as we went along.

But now I'm finding with larger equations and larger numbers, irrational no.s that cannot be represented as a fraction, tat the first method is not easy and can be confusing and leads to longer solutions.

The problem for me mainly is that I have the longer method drummed into me so much that I automatically start using it to write out a solution. Although I am starting to use the 2nd method a lot now I'm doing statistics.

 

[rollcast]

Thanks guys,

In school when we where introduced to algebra we were taught to use the first method of solving various parts as we went along.

But now I'm finding with larger equations and larger numbers, irrational no.s that cannot be represented as a fraction, tat the first method is not easy and can be confusing and leads to longer solutions.

The problem for me mainly is that I have the longer method drummed into me so much that I automatically start using it to write out a solution. Although I am starting to use the 2nd method a lot now I'm doing statistics.

 

[Mentallic]

A lot of questions in maths ask you to simplify an expression as much as possible. You can't do this with any legal calculators in class and the first method you showed doesn't help you get there either.
Use the second method as much as possible as practice for simplifying expressions, which is a really handy skill.

 

[LCKurtz]

Also, your teacher will surely appreciate your not putting numbers in for the variables until the very end. If he is grading an exam or homework problem that is incorrect, he is not likely to punch in all the numbers himself to see where the mistake is. Looking at the algebra he might find that it is just some simple sign error and give you much partial credit. With a solution full of decimals and the wrong answer, just kiss the credit goodbye.

 

[dalcde]

I'd prefer the second since the result can be easily reused for other numbers.