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受欢迎的三角函数 >

8.59tan^2(θ)-24tan(θ)+15.29=0

解答

8.59 tan^{2} (θ) - 24 tan (θ) + 15.29 = 0

解答

θ = 1.06631 \dots + πn, θ = 0.77671 \dots + πn

+1

度数

θ = 61.09560 \dots^{\circ} + 18 0^{\circ} n, θ = 44.50236 \dots^{\circ} + 18 0^{\circ} n

求解步骤

8.59 tan^{2} (θ) - 24 tan (θ) + 15.29 = 0

用替代法求解

8.59 tan^{2} (θ) - 24 tan (θ) + 15.29 = 0

令

: tan (θ) = u

8.59 u^{2} - 24 u + 15.29 = 0

8.59 u^{2} - 24 u + 15.29 = 0 : u = \frac{1200 + 126589}{859}, u = \frac{1200 - 126589}{859}

8.59 u^{2} - 24 u + 15.29 = 0

在两边乘以

100

8.59 u^{2} - 24 u + 15.29 = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

8.59 u^{2} \cdot 100 - 24 u \cdot 100 + 15.29 \cdot 100 = 0 \cdot 100

整理后得

859 u^{2} - 2400 u + 1529 = 0

859 u^{2} - 2400 u + 1529 = 0

使用求根公式求解

859 u^{2} - 2400 u + 1529 = 0

二次方程求根公式:

若

a = 859, b = - 2400, c = 1529

u_{1, 2} = \frac{- ( - 2400 ) \pm ( - 2400 ) ^{2} - 4 \cdot 859 \cdot 1529}{2 \cdot 859}

u_{1, 2} = \frac{- ( - 2400 ) \pm ( - 2400 ) ^{2} - 4 \cdot 859 \cdot 1529}{2 \cdot 859}

(- 2400)^{2} - 4 \cdot 859 \cdot 1529 = 2126589

(- 2400)^{2} - 4 \cdot 859 \cdot 1529

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 2400)^{2} = 240 0^{2}

= 240 0^{2} - 4 \cdot 859 \cdot 1529

数字相乘:

4 \cdot 859 \cdot 1529 = 5253644

= 240 0^{2} - 5253644

240 0^{2} = 5760000

= 5760000 - 5253644

数字相减:

5760000 - 5253644 = 506356

= 506356

506356

质因数分解:

2^{2} \cdot 277 \cdot 457

506356

= 2^{2} \cdot 277 \cdot 457

使用根式运算法则:

n ab = n a n b

= 2^{2} 277 \cdot 457

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2 277 \cdot 457

整理后得

= 2126589

u_{1, 2} = \frac{- ( - 2400 ) \pm 2 126589}{2 \cdot 859}

将解分隔开

u_{1} = \frac{- ( - 2400 ) + 2 126589}{2 \cdot 859}, u_{2} = \frac{- ( - 2400 ) - 2 126589}{2 \cdot 859}

u = \frac{- ( - 2400 ) + 2 126589}{2 \cdot 859} : \frac{1200 + 126589}{859}

\frac{- ( - 2400 ) + 2 126589}{2 \cdot 859}

使用法则

- (- a) = a

= \frac{2400 + 2 126589}{2 \cdot 859}

数字相乘:

2 \cdot 859 = 1718

= \frac{2400 + 2 126589}{1718}

分解

2400 + 2126589 : 2 (1200 + 126589)

2400 + 2126589

改写为

= 2 \cdot 1200 + 2126589

因式分解出通项

2

= 2 (1200 + 126589)

= \frac{2 ( 1200 + 126589 )}{1718}

约分:

2

= \frac{1200 + 126589}{859}

u = \frac{- ( - 2400 ) - 2 126589}{2 \cdot 859} : \frac{1200 - 126589}{859}

\frac{- ( - 2400 ) - 2 126589}{2 \cdot 859}

使用法则

- (- a) = a

= \frac{2400 - 2 126589}{2 \cdot 859}

数字相乘:

2 \cdot 859 = 1718

= \frac{2400 - 2 126589}{1718}

分解

2400 - 2126589 : 2 (1200 - 126589)

2400 - 2126589

改写为

= 2 \cdot 1200 - 2126589

因式分解出通项

2

= 2 (1200 - 126589)

= \frac{2 ( 1200 - 126589 )}{1718}

约分:

2

= \frac{1200 - 126589}{859}

二次方程组的解是:

u = \frac{1200 + 126589}{859}, u = \frac{1200 - 126589}{859}

u = tan (θ)

代回

tan (θ) = \frac{1200 + 126589}{859}, tan (θ) = \frac{1200 - 126589}{859}

tan (θ) = \frac{1200 + 126589}{859}, tan (θ) = \frac{1200 - 126589}{859}

tan (θ) = \frac{1200 + 126589}{859} : θ = arctan (\frac{1200 + 126589}{859}) + πn

tan (θ) = \frac{1200 + 126589}{859}

使用反三角函数性质

tan (θ) = \frac{1200 + 126589}{859}

tan (θ) = \frac{1200 + 126589}{859}

的通解

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

θ = arctan (\frac{1200 + 126589}{859}) + πn

θ = arctan (\frac{1200 + 126589}{859}) + πn

tan (θ) = \frac{1200 - 126589}{859} : θ = arctan (\frac{1200 - 126589}{859}) + πn

tan (θ) = \frac{1200 - 126589}{859}

使用反三角函数性质

tan (θ) = \frac{1200 - 126589}{859}

tan (θ) = \frac{1200 - 126589}{859}

的通解

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

θ = arctan (\frac{1200 - 126589}{859}) + πn

θ = arctan (\frac{1200 - 126589}{859}) + πn

合并所有解

θ = arctan (\frac{1200 + 126589}{859}) + πn, θ = arctan (\frac{1200 - 126589}{859}) + πn

以小数形式表示解

θ = 1.06631 \dots + πn, θ = 0.77671 \dots + πn

作图

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