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sin^2(θ)=(cos^2(θ)-1)/2

Solución

sin^{2} (θ) = \frac{cos ^{2} ( θ ) - 1}{2}

θ = 2 πn, θ = π + 2 πn

Grados

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Pasos de solución

sin^{2} (θ) = \frac{cos ^{2} ( θ ) - 1}{2}

Restar

\frac{cos ^{2} ( θ ) - 1}{2}

de ambos lados

sin^{2} (θ) - \frac{cos ^{2} ( θ ) - 1}{2} = 0

Simplificar

sin^{2} (θ) - \frac{cos ^{2} ( θ ) - 1}{2} : \frac{2 sin ^{2} ( θ ) - cos ^{2} ( θ ) + 1}{2}

sin^{2} (θ) - \frac{cos ^{2} ( θ ) - 1}{2}

Convertir a fracción:

sin^{2} (θ) = \frac{sin ^{2} ( θ ) 2}{2}

= \frac{sin ^{2} ( θ ) \cdot 2}{2} - \frac{cos ^{2} ( θ ) - 1}{2}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{sin ^{2} ( θ ) \cdot 2 - ( cos ^{2} ( θ ) - 1 )}{2}

Expandir

sin^{2} (θ) \cdot 2 - (cos^{2} (θ) - 1) : sin^{2} (θ) \cdot 2 - cos^{2} (θ) + 1

sin^{2} (θ) \cdot 2 - (cos^{2} (θ) - 1)

= 2 sin^{2} (θ) - (cos^{2} (θ) - 1)

- (cos^{2} (θ) - 1) : - cos^{2} (θ) + 1

- (cos^{2} (θ) - 1)

Poner los parentesis

= - (cos^{2} (θ)) - (- 1)

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

= - cos^{2} (θ) + 1

= sin^{2} (θ) \cdot 2 - cos^{2} (θ) + 1

= \frac{2 sin ^{2} ( θ ) - cos ^{2} ( θ ) + 1}{2}

\frac{2 sin ^{2} ( θ ) - cos ^{2} ( θ ) + 1}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 sin^{2} (θ) - cos^{2} (θ) + 1 = 0

Re-escribir usando identidades trigonométricas

1 - cos^{2} (θ) + 2 sin^{2} (θ)

Utilizar la identidad pitagórica:

1 = cos^{2} (x) + sin^{2} (x)

1 - cos^{2} (x) = sin^{2} (x)

= 2 sin^{2} (θ) + sin^{2} (θ)

Simplificar

= 3 sin^{2} (θ)

3 sin^{2} (θ) = 0

Dividir ambos lados entre

3

3 sin^{2} (θ) = 0

Dividir ambos lados entre

3

3 sin^{2} (θ) = 0

Dividir ambos lados entre

3

\frac{3 sin ^{2} ( θ )}{3} = \frac{0}{3}

Simplificar

sin^{2} (θ) = 0

sin^{2} (θ) = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 0

sin (θ) = 0

Soluciones generales para

sin (θ) = 0

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Resolver

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

tan (x) = \frac{3}{10}

y = 1 + sin (\frac{π x}{4}), x

sec (\frac{x}{2}) = 2

arccos (2 x) + arccos (x) = \frac{π}{3}

tan (2 x) = \frac{500}{350 - 75}