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Popular Trigonometría >

cos(A-25)+cos(A+25)=cos(25)

Solución

cos (A - 2 5^{\circ}) + cos (A + 2 5^{\circ}) = cos (2 5^{\circ})

Solución

A = 6 0^{\circ} + 36 0^{\circ} n, A = 30 0^{\circ} + 36 0^{\circ} n

Radianes

A = \frac{π}{3} + 2 πn, A = \frac{5 π}{3} + 2 πn

Pasos de solución

cos (A - 2 5^{\circ}) + cos (A + 2 5^{\circ}) = cos (2 5^{\circ})

Re-escribir usando identidades trigonométricas

cos (A - 2 5^{\circ}) + cos (A + 2 5^{\circ})

Utilizar la identidad suma-producto:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

= 2 cos (\frac{A - 2 5 ^{\circ} + A + 2 5 ^{\circ}}{2}) cos (\frac{A - 2 5 ^{\circ} - ( A + 2 5 ^{\circ} )}{2})

Simplificar

2 cos (\frac{A - 2 5 ^{\circ} + A + 2 5 ^{\circ}}{2}) cos (\frac{A - 2 5 ^{\circ} - ( A + 2 5 ^{\circ} )}{2}) : 2 cos (2 5^{\circ}) cos (A)

2 cos (\frac{A - 2 5 ^{\circ} + A + 2 5 ^{\circ}}{2}) cos (\frac{A - 2 5 ^{\circ} - ( A + 2 5 ^{\circ} )}{2})

\frac{A - 2 5 ^{\circ} + A + 2 5 ^{\circ}}{2} = A

\frac{A - 2 5 ^{\circ} + A + 2 5 ^{\circ}}{2}

A - 2 5^{\circ} + A + 2 5^{\circ} = 2 A

A - 2 5^{\circ} + A + 2 5^{\circ}

Agrupar términos semejantes

= A + A - 2 5^{\circ} + 2 5^{\circ}

Sumar elementos similares:

A + A = 2 A

= 2 A - 2 5^{\circ} + 2 5^{\circ}

Sumar elementos similares:

- 2 5^{\circ} + 2 5^{\circ} = 0

= 2 A

= \frac{2 A}{2}

Dividir:

\frac{2}{2} = 1

= A

= 2 cos (A) cos (\frac{A - ( A + 2 5 ^{\circ} ) - 2 5 ^{\circ}}{2})

\frac{A - 2 5 ^{\circ} - ( A + 2 5 ^{\circ} )}{2} = - 2 5^{\circ}

\frac{A - 2 5 ^{\circ} - ( A + 2 5 ^{\circ} )}{2}

Simplificar

A - 2 5^{\circ} - (A + 2 5^{\circ})

en una fracción:

- 5 0^{\circ}

A - 2 5^{\circ} - (A + 2 5^{\circ})

Convertir a fracción:

A = \frac{A 36}{36}, (A + 2 5^{\circ}) = \frac{( A + 2 5 ^{\circ} ) 36}{36}

= \frac{A \cdot 36}{36} - 2 5^{\circ} - \frac{( A + 2 5 ^{\circ} ) \cdot 36}{36}

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{A \cdot 36 - 90 0 ^{\circ} - ( A + 2 5 ^{\circ} ) \cdot 36}{36}

Expandir

A \cdot 36 - 90 0^{\circ} - (A + 2 5^{\circ}) \cdot 36 : - 180 0^{\circ}

A \cdot 36 - 90 0^{\circ} - (A + 2 5^{\circ}) \cdot 36

= 36 A - 90 0^{\circ} - 36 (A + 2 5^{\circ})

Expandir

- 36 (A + 2 5^{\circ}) : - 36 A - 90 0^{\circ}

- 36 (A + 2 5^{\circ})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = - 36, b = A, c = 2 5^{\circ}

= - 36 A + (- 36) 2 5^{\circ}

Aplicar las reglas de los signos

+ (- a) = - a

= - 36 A - 36 \cdot 2 5^{\circ}

36 \cdot 2 5^{\circ} = 90 0^{\circ}

36 \cdot 2 5^{\circ}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 90 0^{\circ}

Eliminar los terminos comunes:

36

= 90 0^{\circ}

= - 36 A - 90 0^{\circ}

= A \cdot 36 - 90 0^{\circ} - 36 A - 90 0^{\circ}

Simplificar

A \cdot 36 - 90 0^{\circ} - 36 A - 90 0^{\circ} : - 180 0^{\circ}

A \cdot 36 - 90 0^{\circ} - 36 A - 90 0^{\circ}

Agrupar términos semejantes

= 36 A - 36 A - 90 0^{\circ} - 90 0^{\circ}

Sumar elementos similares:

36 A - 36 A = 0

= - 90 0^{\circ} - 90 0^{\circ}

Sumar elementos similares:

- 90 0^{\circ} - 90 0^{\circ} = - 180 0^{\circ}

= - 180 0^{\circ}

= - 180 0^{\circ}

= \frac{- 180 0 ^{\circ}}{36}

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

= - 5 0^{\circ}

Eliminar los terminos comunes:

2

= - 5 0^{\circ}

= \frac{- 5 0 ^{\circ}}{2}

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{5 0 ^{\circ}}{2}

Aplicar las propiedades de las fracciones:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

\frac{5 0 ^{\circ}}{2} = \frac{90 0 ^{\circ}}{18 \cdot 2}

= - \frac{90 0 ^{\circ}}{18 \cdot 2}

Multiplicar los numeros:

18 \cdot 2 = 36

= - 2 5^{\circ}

= 2 cos (- 2 5^{\circ}) cos (A)

Simplificar

cos (- 2 5^{\circ}) : cos (2 5^{\circ})

cos (- 2 5^{\circ})

Utilizar la siguiente propiedad:

cos (- x) = cos (x)

cos (- 2 5^{\circ}) = cos (2 5^{\circ})

= cos (2 5^{\circ})

= 2 cos (2 5^{\circ}) cos (A)

= 2 cos (2 5^{\circ}) cos (A)

2 cos (2 5^{\circ}) cos (A) = cos (2 5^{\circ})

Dividir ambos lados entre

2 cos (2 5^{\circ})

2 cos (2 5^{\circ}) cos (A) = cos (2 5^{\circ})

Dividir ambos lados entre

2 cos (2 5^{\circ})

\frac{2 cos ( 2 5 ^{\circ} ) cos ( A )}{2 cos ( 2 5 ^{\circ} )} = \frac{cos ( 2 5 ^{\circ} )}{2 cos ( 2 5 ^{\circ} )}

Simplificar

cos (A) = \frac{1}{2}

cos (A) = \frac{1}{2}

Soluciones generales para

cos (A) = \frac{1}{2}

cos (x)

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

A = 6 0^{\circ} + 36 0^{\circ} n, A = 30 0^{\circ} + 36 0^{\circ} n

A = 6 0^{\circ} + 36 0^{\circ} n, A = 30 0^{\circ} + 36 0^{\circ} n

Gráfica

Ejemplos populares

tan^2(θ)-tan(θ)=0,θ,0<= θ<2pi