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受欢迎的三角函数 >

3tan^2(θ)-1=0

解答

3 tan^{2} (θ) - 1 = 0

解答

θ = 0.52359 \dots + πn, θ = - 0.52359 \dots + πn

+1

度数

θ = 3 0^{\circ} + 18 0^{\circ} n, θ = - 3 0^{\circ} + 18 0^{\circ} n

求解步骤

3 tan^{2} (θ) - 1 = 0

用替代法求解

3 tan^{2} (θ) - 1 = 0

令

: tan (θ) = u

3 u^{2} - 1 = 0

3 u^{2} - 1 = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

3 u^{2} - 1 = 0

将

1

到右边

3 u^{2} - 1 = 0

两边加上

1

3 u^{2} - 1 + 1 = 0 + 1

化简

3 u^{2} = 1

3 u^{2} = 1

两边除以

3

3 u^{2} = 1

两边除以

3

\frac{3 u ^{2}}{3} = \frac{1}{3}

化简

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{1}{3}, u = - \frac{1}{3}

u = tan (θ)

代回

tan (θ) = \frac{1}{3}, tan (θ) = - \frac{1}{3}

tan (θ) = \frac{1}{3}, tan (θ) = - \frac{1}{3}

tan (θ) = \frac{1}{3} : θ = arctan (\frac{1}{3}) + πn

tan (θ) = \frac{1}{3}

使用反三角函数性质

tan (θ) = \frac{1}{3}

tan (θ) = \frac{1}{3}

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

θ = arctan (\frac{1}{3}) + πn

θ = arctan (\frac{1}{3}) + πn

tan (θ) = - \frac{1}{3} : θ = arctan (- \frac{1}{3}) + πn

tan (θ) = - \frac{1}{3}

使用反三角函数性质

tan (θ) = - \frac{1}{3}

tan (θ) = - \frac{1}{3}

的通解

tan (x) = - a \Rightarrow x = arctan (- a) + πn

θ = arctan (- \frac{1}{3}) + πn

θ = arctan (- \frac{1}{3}) + πn

合并所有解

θ = arctan (\frac{1}{3}) + πn, θ = arctan (- \frac{1}{3}) + πn

以小数形式表示解

θ = 0.52359 \dots + πn, θ = - 0.52359 \dots + πn

作图

流行的例子

sin(2θ-pi/2)=-1

sin (2 θ - \frac{π}{2}) = - 1

sin(x/2)+cos(x)=0

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