Mathematic Algebras

Algebra is a method to simplify calculation, a method to reunite broken parts, a method to solve puzzles. Algebra skills are very important for an electronic engineer (or maybe for hackers) to solve any circuit problems related to mathematic. Not too much words, below is example of algebra problems and the problem solver:

Algebra Problem 1. simplify :

4a + 3b² – a + 2 = 38

Problem Solving:

4a + 3b² – a + 2 = 38

(4a – a) + 3b² = 38 – 2

3a + 3b² = 36

√3a + 3b = 6

Algebra Problem 2, find y :

5y+50 = 2 (3y + 10 )

Problem solving:

5y+50 = 2 (3y + 10 )

5y+50 = 6y + 20

5y +50 – 20 = 6y

5y + 30 = 6y

30 = 6y – 5y

y = 30

Example problem 3, find value of i:

2 = i + √3

Problem Solving:

2² = i² + 3

4 = i² + 3

i = 1

Example problem 4:

Simplify: 2x + 3(x + 10) = 55

Problem Solving

2x + 3x + 30 = 55

5x = (55-30)

5x = 25

x = 5

 

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Basic Trigonemetry for Robotic Mechanical Joints

Trigonometry is a branch of mathematic which frequently used in robotics where it’s commonly used in inverse kinematics of mechanical leg / joints, forward kinematics, etc. Trigonometry calculates about angles and sides of a triangle. In trigonemetric, there’s ratios for acute angles:

– sin = opposite  /  hypotenuse

– cos = adjacent / hypotenuse

– tan = opposite  / adjacent

– secant = hypotenuse / adjacent

– cosecant = hypotenuse / opposite

– cotangent = adjacent / opposite

For example here we have a triangle:

it’s known that adjacent’s length is 4 and hypotenuse’s length is 5. Based on phytagoras theorem:

 

So :

4²  + b² = 5²

16 + b² = 25

b = √9 = 3

So we got :

Sin R = b / c = 3 / 5, Cos R = a / c = 4 / 5, Tan R = b/a = 3/4, Secant R = c/a = 5/4, Cosecant R = c  / b =5/3, cotangent R = a/b = 4/3

Trigonometry on Equilateral Triangle

For example here, we have an equilateral triangle:

Actually that equilateral triangle can be divide into 2 area, something like this :

That equilateral triangle when divided into 2 areas, has each 30 degress for it’s top.

Let’s just give name for each angle on cartesius coordinate as x and y:

How to calculate sin x, cos x , tan x, sin y, cos y and tan y ?

Have a check on special table of trigonometry :

Since sin x is 30 degree = 1/2, where sin = opposite  /  hypotenuse, so sin x = b / c, we figure out that b = 1 and c = 2. To find a based on phytagoras theory:

c² = a² + b²

2² = a² + 1²

4 = a² +  1

a = √3

So cos x = adjacent / hypotenuse  = a / c = √3 / 2

tan x = opposite  / adjacent = b / a = 1/√3

The same goes for sin, cos and tan y.