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受欢迎的三角函数 >

solvefor θ,1+cos(θ)=3cos(θ)

解答

求解

θ, 1 + cos (θ) = 3 cos (θ)

解答

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

+1

度数

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

求解步骤

1 + cos (θ) = 3 cos (θ)

用替代法求解

1 + cos (θ) = 3 cos (θ)

令

: cos (θ) = u

1 + u = 3 u

1 + u = 3 u : u = \frac{1}{2}

1 + u = 3 u

将

1

到右边

1 + u = 3 u

两边减去

1

1 + u - 1 = 3 u - 1

化简

u = 3 u - 1

u = 3 u - 1

将

3 u

para o lado esquerdo

u = 3 u - 1

两边减去

3 u

u - 3 u = 3 u - 1 - 3 u

化简

- 2 u = - 1

- 2 u = - 1

两边除以

- 2

- 2 u = - 1

两边除以

- 2

\frac{- 2 u}{- 2} = \frac{- 1}{- 2}

化简

u = \frac{1}{2}

u = \frac{1}{2}

u = cos (θ)

代回

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2} : θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

的通解

cos (x)

周期表（周期为

2 πn

）：

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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

合并所有解

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

作图

流行的例子

3sin(x)-4cos(x)=0

3 sin (x) - 4 cos (x) = 0

sin(θ)= 1/4 ,0<θ<90

sin (θ) = \frac{1}{4}, 0^{\circ} < θ < 9 0^{\circ}

sin(90-2x)=cos(90-x)

sin (90 - 2 x) = cos (9 0^{\circ} - x)

tan (\frac{x}{2}) = - 0.39

4 sin (3 x) - 1 = 0