- #1
andycampbell1
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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[tex]\alpha[/tex] = 5[tex]\pi[/tex]/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([tex]\omega[/tex]t+[tex]\alpha[/tex])
Homework Equations
The Attempt at a Solution
The solution is
y=sin t -cos t = A cos([tex]\omega[/tex]t+[tex]\alpha[/tex])
= A cos t cos [tex]\alpha[/tex] - A sin t sin [tex]\alpha[/tex]
A cos [tex]\alpha[/tex] = -1
- A sin [tex]\alpha[/tex] = 1
Squaring the above equations and adding side by side implies that A = [tex]\sqrt{2}[/tex]
Dividing the above equations side by side gives
tan [tex]\alpha[/tex]= 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 [tex]\alpha[/tex] = [tex]\pi[/tex]tan-1 = 5[tex]\pi[/tex]/4
y= sin t - cos t = [tex]\sqrt{2}[/tex]cos(t+ 5[tex]\pi[/tex]/4)
What I do not understand is where did 5[tex]\pi[/tex] come from? I have checked my notes and it does not mention anything about where pi comes from.