Sen x/2 + cos x/2 = 1
Hola
sen(x/2) + cos(x/2)
Dividimos todo por √2
(sen(x/2))/√2 + (cos(x/2))√2 = 1/√2
Presentamos distinto
(1/√2) sen(x/2) + (1/√2) cos(x/2) = 1/√2
recordamos que
sen(45º) = cos(45º) = 1/√2
cos(45º) sen(x/2) + sen(45º) cos(x/2) = sen(45º)
sen(45º + (x/2)) = sen(45º)
Dos caminos
Primer camino
45º + (x/2) = 45º + 360º*k
k: cualquier entero
(x/2) = 360º * k
x = 720º * k
****************
Segundo camino
45º + (x/2) + 45º = 180º + 360º*k
(x/2) + 90º = 180º + 360º*k
(x/2) = 90º + 360º*k
x =180º + 720º * k
**********************
Do you know this identity?
sin(a + b) = sin(a).cos(b) + cos(a).sin(b) → suppose that: b = φ
sin(a + φ) = sin(a).cos(φ) + cos(a).sin(φ) → you multiply by C
C.sin(a + φ) = C.sin(a).cos(φ) + C.cos(a).sin(φ) → let: C.cos(φ) = 1 ← equation (1)
C.sin(a + φ) = sin(a) + C.cos(a).sin(φ) → let: C.sin(φ) = 1 ← equation (2)
C.sin(a + φ) = sin(a) + cos(a) → suppose that: a = x/2
C.sin[(x/2) + φ] = sin(x/2) + cos(x/2) ← this is your expression
You can get a system of 2 equations:
(1) : C.cos(φ) = 1
(2) : C.sin(φ) = 1
You calculate (2)/(1)
[C.sin(φ)]/[C.cos(φ)] = 1
sin(φ)/cos(φ) = 1
tan(φ) = 1
φ = π/4
You calculate (1)² + (2)²
[C.cos(φ)]² + [C.sin(φ)]² = 1² + 1²
C².cos²(φ) + C².sin²(φ) = 2
C².[cos²(φ) + sin²(φ)] = 2 → recall the famous formula: cos²(φ) + sin²(φ) = 1
C² = 2
C = √2
Recall your expression:
sin(x/2) + cos(x/2) = C.sin[(x/2) + φ] → where: C = √2
sin(x/2) + cos(x/2) = √2 * sin[(x/2) + φ] → where: φ = π/4
sin(x/2) + cos(x/2) = √2 * sin[(x/2) + (π/4)]
Recall your equation:
sin(x/2) + cos(x/2) = 1 → recall the previous result
√2 * sin[(x/2) + (π/4)] = 1
sin[(x/2) + (π/4)] = 1/√2
sin[(x/2) + (π/4)] = (√2)/2 ← the angle is (π/4)
(x/2) + (π/4) = (π/4) + k2π
x/2 = k2π
x = k4π ← where k is an integer
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así agradecemos a los colaboradores y usuarios
que se preocuparon por responder tus preguntas
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Answers & Comments
Hola
sen(x/2) + cos(x/2)
Dividimos todo por √2
(sen(x/2))/√2 + (cos(x/2))√2 = 1/√2
Presentamos distinto
(1/√2) sen(x/2) + (1/√2) cos(x/2) = 1/√2
recordamos que
sen(45º) = cos(45º) = 1/√2
cos(45º) sen(x/2) + sen(45º) cos(x/2) = sen(45º)
sen(45º + (x/2)) = sen(45º)
Dos caminos
Primer camino
45º + (x/2) = 45º + 360º*k
k: cualquier entero
(x/2) = 360º * k
x = 720º * k
****************
Segundo camino
45º + (x/2) + 45º = 180º + 360º*k
k: cualquier entero
(x/2) + 90º = 180º + 360º*k
(x/2) = 90º + 360º*k
x =180º + 720º * k
**********************
Do you know this identity?
sin(a + b) = sin(a).cos(b) + cos(a).sin(b) → suppose that: b = φ
sin(a + φ) = sin(a).cos(φ) + cos(a).sin(φ) → you multiply by C
C.sin(a + φ) = C.sin(a).cos(φ) + C.cos(a).sin(φ) → let: C.cos(φ) = 1 ← equation (1)
C.sin(a + φ) = sin(a) + C.cos(a).sin(φ) → let: C.sin(φ) = 1 ← equation (2)
C.sin(a + φ) = sin(a) + cos(a) → suppose that: a = x/2
C.sin[(x/2) + φ] = sin(x/2) + cos(x/2) ← this is your expression
You can get a system of 2 equations:
(1) : C.cos(φ) = 1
(2) : C.sin(φ) = 1
You calculate (2)/(1)
[C.sin(φ)]/[C.cos(φ)] = 1
sin(φ)/cos(φ) = 1
tan(φ) = 1
φ = π/4
You calculate (1)² + (2)²
[C.cos(φ)]² + [C.sin(φ)]² = 1² + 1²
C².cos²(φ) + C².sin²(φ) = 2
C².[cos²(φ) + sin²(φ)] = 2 → recall the famous formula: cos²(φ) + sin²(φ) = 1
C² = 2
C = √2
Recall your expression:
sin(x/2) + cos(x/2) = C.sin[(x/2) + φ] → where: C = √2
sin(x/2) + cos(x/2) = √2 * sin[(x/2) + φ] → where: φ = π/4
sin(x/2) + cos(x/2) = √2 * sin[(x/2) + (π/4)]
Recall your equation:
sin(x/2) + cos(x/2) = 1 → recall the previous result
√2 * sin[(x/2) + (π/4)] = 1
sin[(x/2) + (π/4)] = 1/√2
sin[(x/2) + (π/4)] = (√2)/2 ← the angle is (π/4)
(x/2) + (π/4) = (π/4) + k2π
x/2 = k2π
x = k4π ← where k is an integer
Do you know this identity?
sin(a + b) = sin(a).cos(b) + cos(a).sin(b) → suppose that: b = φ
sin(a + φ) = sin(a).cos(φ) + cos(a).sin(φ) → you multiply by C
C.sin(a + φ) = C.sin(a).cos(φ) + C.cos(a).sin(φ) → let: C.cos(φ) = 1 ← equation (1)
C.sin(a + φ) = sin(a) + C.cos(a).sin(φ) → let: C.sin(φ) = 1 ← equation (2)
C.sin(a + φ) = sin(a) + cos(a) → suppose that: a = x/2
C.sin[(x/2) + φ] = sin(x/2) + cos(x/2) ← this is your expression
You can get a system of 2 equations:
(1) : C.cos(φ) = 1
(2) : C.sin(φ) = 1
You calculate (2)/(1)
[C.sin(φ)]/[C.cos(φ)] = 1
sin(φ)/cos(φ) = 1
tan(φ) = 1
φ = π/4
You calculate (1)² + (2)²
[C.cos(φ)]² + [C.sin(φ)]² = 1² + 1²
C².cos²(φ) + C².sin²(φ) = 2
C².[cos²(φ) + sin²(φ)] = 2 → recall the famous formula: cos²(φ) + sin²(φ) = 1
C² = 2
C = √2
Recall your expression:
sin(x/2) + cos(x/2) = C.sin[(x/2) + φ] → where: C = √2
sin(x/2) + cos(x/2) = √2 * sin[(x/2) + φ] → where: φ = π/4
sin(x/2) + cos(x/2) = √2 * sin[(x/2) + (π/4)]
Recall your equation:
sin(x/2) + cos(x/2) = 1 → recall the previous result
√2 * sin[(x/2) + (π/4)] = 1
sin[(x/2) + (π/4)] = 1/√2
sin[(x/2) + (π/4)] = (√2)/2 ← the angle is (π/4)
(x/2) + (π/4) = (π/4) + k2π
x/2 = k2π
x = k4π ← where k is an integer
No olvides elegir una respuesta como la mejor,
así agradecemos a los colaboradores y usuarios
que se preocuparon por responder tus preguntas