# Angle between two tangent lines

1. Jan 20, 2012

### jsmith613

1. The problem statement, all variables and given/known data

Question 8(b)

2. Relevant equations

3. The attempt at a solution

Ok so I found both values of dy/dx for BOTH EQUATIONS

y = x2 - 4x → 2x - 4
y = |4x - x2| → |4 - 2x| could someone please claify this, I have never differentiated a modulus before

Thus the two gradients of the lines are 4 and -4 BUT HOW DO I GO ON TO FIND THE angle between the tangents??

Last edited by a moderator: May 5, 2017
2. Jan 20, 2012

### HallsofIvy

$|4x- x^2|= |x(4- x)|$ and is equal to $4x- x^2$ for x between 0 and 4 but equal to $x^2- 4x$ for x< 0 or x> 4. That is, its derivative if 4- 2x for x between 0 and 4 and equal to 2x- 4 for x< 0 or x> 4. At x= 4, there is a "cusp" so technically, there is no derivative. Of course, you are interested in the curve between 0 and 4 so you really want $\lim_{x\to 4^-} 4- 2x= -4$ as you say.

To find the angle between the lines remember that the derivative is the tangent of the angle the curve makes with the horizontal. And that
$$tan(\theta- \phi)= \frac{tan(\theta)- tan(\phi)}{1+ tan(\theta)tan(\phi)}$$

3. Jan 21, 2012

### jsmith613

I never knew this
is this A-level maths or beyond?
could you please explain why this is true?

4. Jan 21, 2012

### SammyS

Staff Emeritus
It's usually taught in Calculus when you first learn about derivatives representing the slope of the tangent line.

It's often taught in trigonometry that the slope of a line is equal to the tangent of the angle the line makes with the x-axis.

5. Jan 21, 2012

### jsmith613

thanks for this :)
I will note this rule!!