Trigonometry Engineering Waves proble

[andycampbell1]

Homework Statement

Hi

I am doing a maths past paper and I would like some help to understand where part of the solution came from. The part I do not understand is the bit at the end which says tan$$\alpha$$ = 5$$\pi$$/4. It would be great if someone could help me understand where this came from.

The question is

y = sin t - cos t in the form A cos($$\omega$$t+$$\alpha$$)

Homework Equations

The Attempt at a Solution

The solution is

y=sin t -cos t = A cos($$\omega$$t+$$\alpha$$)
= A cos t cos $$\alpha$$ - A sin t sin $$\alpha$$

A cos $$\alpha$$ = -1
- A sin $$\alpha$$ = 1

Squaring the above equations and adding side by side implies that A = $$\sqrt{2}$$

Dividing the above equations side by side gives

tan $$\alpha$$= 1

tan is positive in the first and third quadrant. We choose the third quadrant since from the above equation sin and cos are negative.

Hence $$\alpha$$ = $$\pi$$tan^-1 = 5$$\pi$$/4

y= sin t - cos t = $$\sqrt{2}$$cos(t+ 5$$\pi$$/4)

What I do not understand is where did 5$$\pi$$ come from? I have checked my notes and it does not mention anything about where pi comes from.

 

[tiny-tim]

hi andy!

(have an alpha: α and a pi: π and an omega: ω and a square-root: √ )

tan(π/4) = tan(45°) = 1

tan(π + x) = tanx, so tan(π + π/4) = 1 also, ie tan(5π/4) = 1

 

[andycampbell1]

Thanks. If I was using the cast rule does each quadrant have its own value of pi? I get the first part but I don't get the second line. Sorry ha I'm just hopeless at trigonometry.

 

[atomthick]

Thanks. If I was using the cast rule does each quadrant have its own value of pi? I get the first part but I don't get the second line. Sorry ha I'm just hopeless at trigonometry.

You just said you have chosen the third quadrant therefore pi + pi/4 is the value you are looking for because it's in the third quadrant.

 

[andycampbell1]

I know how to work out what quadrant to put tan in but. I do not know what each quadrant represents in terms of pi. Like u said pi in the 3rd quadrant = pi
+ pi/4. What would the other quadrants wqual

 

[andycampbell1]

I know how to work out what quadrant to put tan in but. I do not know what each quadrant represents in terms of pi. Like u said pi in the 3rd quadrant = pi
+ pi/4. What would the other quadrants equal? The solution I posted came with the past paper I am just trying to make sense of it.

 

[atomthick]

Each quadrant has 90^o (or $$\pi/2$$ radians). The quadrant ranges are:

First quadrant ranges from 0 to $$\pi/2$$
The second from $$\pi/2$$ to $$\pi$$
The third from $$\pi$$ to $$3\pi/2$$
The fourth from $$3\pi/2$$ to $$2\pi$$

 

[andycampbell1]

Thanks, from the examples I have the quadrant values make them make sense to me now, thanks for your help.