Trig Equation?
Solve for x.

sin x + cos x = 0
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answers:
Captain Matticus, LandPiratesInc say: sin(x) + cos(x) = 0
sin(x) = -cos(x)
sin(x)/cos(x) = -1
tan(x) = -1
x = 3pi/4 + pi * k

k is an integer


sin(x) + cos(x) = 0
1 * sin(x) + 1 * cos(x) = 0
sqrt(1^2 + 1^2) * (sin(x) * 1 / sqrt(1^2 + 1^2) + cos(x) * 1 / sqrt(1^2 + 1^2)) = 0
sqrt(2) * (sin(x) * sqrt(2)/2 + cos(x) * sqrt(2)/2) = 0
sqrt(2) * (sin(x) * cos(pi/4) + cos(x) * sin(pi/4)) = 0
sqrt(2) * sin(x + pi/4) = 0
sin(x + pi/4) = 0
x + pi/4 = pi * k
x = -pi/4 + pi * k

Which corresponds with 3pi/4 + pi * k

sqrt(2) * (sin(x) * sqrt(2)/2 + cos(x) * sqrt(2)/2) = 0
sqrt(2) * (sin(x) * sin(pi/4) + cos(x) * cos(pi/4)) = 0
sqrt(2) * cos(x - pi/4) = 0
cos(x - pi/4) = 0
x - pi/4 = pi/2 + pi * k
x = 3pi/4 + pi * k

Same answer, 3 different methods
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MyRank say: sinx + cosx = 0

sinx = -cosx

sinx/cox = -1

tanx = -1

x = tan⁻¹(-1)

x = -tan⁻¹(tanπ/4)

x = -π/4 (or) π + π/4.
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Como say: sin x + cos x = k cos ( x - a )
sin x + cos x = k cos a cos x + k sin a sin x
1 = k cos a
1 = k sin a

tan a = 1
a = π/4

1² + 1² = k²
k = √2

√2 cos ( x - π/4 ) = 0
x - π/4 = π/2 , 3π/2
x = 3π/4 , 7π/4
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lenpol7 say: Sinx = -Cosx
Sinx / Cosx = -1
Tan x = -1
x = Tan^-1(-1)
x = -45 degrees, 135 degrees. 315 degrees.
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oldprof say: sinx/cosx = - 1 = tanx; so x = ATAN(-1 ) = -45 deg below the XX axis.
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alex say: sin x + cos x = 0 --->tanx = -1 --->x = -π/4 + kπ
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khalil say: sin x + cos x = 0
cos x = sin ( π/2 - x)
sin x + sin (π/2 - x) = 0
sin α + sin ß = 2 sin (α+ß)/2 * cos (α-ß)/2
2 sin π/4 cos ( x - π/4) = 0
cos (x - π/4) = 0 = cos π/2
1)
x - π/4 = π/2 + 2nπ ... n is an integer ..x = 3π/4 + 2nπ ◄◄◄
2)
x - π/4 = 2π - π/2 + 2nπ ..... x = 7π/4 + 2nπ ◄◄◄
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sepia say: sin(x) + cos(x) = 0
sin(x) = -cos(x)
Solution:
x ≈ 0.25000 (12.566 n - 3.1416), n element Z
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Johnathan say: sin(x) + cos(x) = 0

(sin(x) + cos(x))(sin(x) - cos(x)) = 0(sin(x) - cos(x))

sin^2(x) - cos^2(x) = 0

sin^2(x) - (1 - sin^2(x)) = 0

2 sin^2(x) - 1 = 0

2 sin^2(x) = 1

sin^2(x) = 1/2

sin(x) = +/- sqrt(1/2) = +/- 1/sqrt(2) -> x = pi/4 +(n/2)pi, if n is an integer.

But wait a minute...!!

This won't work in quadrants 1 and 3 because sin(x) and cos(x) SHARE THE SAME SIGN. For the equation to be satisfied, they must DIFFER in sign -- in other words, x = 3pi/4 + n(pi), again if n is an integer.
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