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受欢迎的三角函数 >

(tan^2(θ)-9)(2cos(θ)+1)=0

解答

(tan^{2} (θ) - 9) (2 cos (θ) + 1) = 0

解答

θ = 1.24904 \dots + πn, θ = - 1.24904 \dots + πn, θ = \frac{2 π}{3} + 2 πn, θ = \frac{4 π}{3} + 2 πn

+1

度数

θ = 71.56505 \dots^{\circ} + 18 0^{\circ} n, θ = - 71.56505 \dots^{\circ} + 18 0^{\circ} n, θ = 12 0^{\circ} + 36 0^{\circ} n, θ = 24 0^{\circ} + 36 0^{\circ} n

求解步骤

(tan^{2} (θ) - 9) (2 cos (θ) + 1) = 0

分别求解每个部分

tan^{2} (θ) - 9 = 0 or 2 cos (θ) + 1 = 0

tan^{2} (θ) - 9 = 0 : θ = arctan (3) + πn, θ = arctan (- 3) + πn

tan^{2} (θ) - 9 = 0

用替代法求解

tan^{2} (θ) - 9 = 0

令

: tan (θ) = u

u^{2} - 9 = 0

u^{2} - 9 = 0 : u = 3, u = - 3

u^{2} - 9 = 0

将

9

到右边

u^{2} - 9 = 0

两边加上

9

u^{2} - 9 + 9 = 0 + 9

化简

u^{2} = 9

u^{2} = 9

对于

x^{2} = f (a)

解为

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

u = 9, u = - 9

9 = 3

9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 3

- 9 = - 3

- 9

9 = 3

9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 3

= - 3

u = 3, u = - 3

u = tan (θ)

代回

tan (θ) = 3, tan (θ) = - 3

tan (θ) = 3, tan (θ) = - 3

tan (θ) = 3 : θ = arctan (3) + πn

tan (θ) = 3

使用反三角函数性质

tan (θ) = 3

tan (θ) = 3

的通解

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

θ = arctan (3) + πn

θ = arctan (3) + πn

tan (θ) = - 3 : θ = arctan (- 3) + πn

tan (θ) = - 3

使用反三角函数性质

tan (θ) = - 3

tan (θ) = - 3

的通解

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

θ = arctan (- 3) + πn

θ = arctan (- 3) + πn

合并所有解

θ = arctan (3) + πn, θ = arctan (- 3) + πn

2 cos (θ) + 1 = 0 : θ = \frac{2 π}{3} + 2 πn, θ = \frac{4 π}{3} + 2 πn

2 cos (θ) + 1 = 0

将

1

到右边

2 cos (θ) + 1 = 0

两边减去

1

2 cos (θ) + 1 - 1 = 0 - 1

化简

2 cos (θ) = - 1

2 cos (θ) = - 1

两边除以

2

2 cos (θ) = - 1

两边除以

2

\frac{2 cos ( θ )}{2} = \frac{- 1}{2}

化简

cos (θ) = - \frac{1}{2}

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{2 π}{3} + 2 πn, θ = \frac{4 π}{3} + 2 πn

θ = \frac{2 π}{3} + 2 πn, θ = \frac{4 π}{3} + 2 πn

合并所有解

θ = arctan (3) + πn, θ = arctan (- 3) + πn, θ = \frac{2 π}{3} + 2 πn, θ = \frac{4 π}{3} + 2 πn

以小数形式表示解

θ = 1.24904 \dots + πn, θ = - 1.24904 \dots + πn, θ = \frac{2 π}{3} + 2 πn, θ = \frac{4 π}{3} + 2 πn

作图

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