# math

sinA=.....+cosA

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1. I did this yesterday

http://www.jiskha.com/display.cgi?id=1459511832

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posted by Reiny

sinA = x + cosA
x = sinA - cosA

let's change sinA - cosA into a single sinusoidal expression:

Let sinA - cosA = a sin(A + k)

a sin(A+k)
= a(sinAcosk + cosAsink_
= asinAcosk + acosAsink

then asinAcosk = sinA
acosk = 1
cosk = 1/a

acosAsink = -cosA
asink = -1
sink = -1/a

but we know sin^2 k + cos^2 k = 1
(-1/a)^2 + (1/a)^2 = 1
2/a^2 = 1
a^2 = 2
a = sqrt(2)

also tank = sink/cosk = (-1/a) / (1/a) = -1
k = -Pi/4

so sinA - cosA = sqrt(2) sin(A - Pi/4)

x = sqrt(2) sin(A - Pi/4)

so I plotted on Wolfram:
y = sinx - cosx and y = sqrt(2) sin(x - Pi/4)
and the two curves coincides

https://www.wolframalpha.com/input/?i=plot+y+%3D+sinx+-+cosx,+y+%3D+sqrt(2)+sin(x+-+Pi%2F4)

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posted by Reiny

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