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受欢迎的三角函数 >

10=6.7+13.75tan(θ)-4.718(1+tan^2(θ))

解答

10 = 6.7 + 13.75 tan (θ) - 4.718 (1 + tan^{2} (θ))

解答

θ = 0.67843 \dots + πn, θ = 1.12790 \dots + πn

+1

度数

θ = 38.87154 \dots^{\circ} + 18 0^{\circ} n, θ = 64.62419 \dots^{\circ} + 18 0^{\circ} n

求解步骤

10 = 6.7 + 13.75 tan (θ) - 4.718 (1 + tan^{2} (θ))

交换两边

6.7 + 13.75 tan (θ) - 4.718 (1 + tan^{2} (θ)) = 10

用替代法求解

6.7 + 13.75 tan (θ) - 4.718 (1 + tan^{2} (θ)) = 10

令

: tan (θ) = u

6.7 + 13.75 u - 4.718 (1 + u^{2}) = 10

6.7 + 13.75 u - 4.718 (1 + u^{2}) = 10 : u = \frac{6875 - 9436701}{4718}, u = \frac{6875 + 9436701}{4718}

6.7 + 13.75 u - 4.718 (1 + u^{2}) = 10

在两边乘以

1000

6.7 + 13.75 u - 4.718 (1 + u^{2}) = 10

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

3

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

1000

6.7 \cdot 1000 + 13.75 u \cdot 1000 - 4.718 (1 + u^{2}) \cdot 1000 = 10 \cdot 1000

整理后得

6700 + 13750 u - 4718 (1 + u^{2}) = 10000

6700 + 13750 u - 4718 (1 + u^{2}) = 10000

展开

6700 + 13750 u - 4718 (1 + u^{2}) : - 4718 u^{2} + 13750 u + 1982

6700 + 13750 u - 4718 (1 + u^{2})

乘开

- 4718 (1 + u^{2}) : - 4718 - 4718 u^{2}

- 4718 (1 + u^{2})

使用分配律:

a (b + c) = ab + a c

a = - 4718, b = 1, c = u^{2}

= - 4718 \cdot 1 + (- 4718) u^{2}

使用加减运算法则

+ (- a) = - a

= - 4718 \cdot 1 - 4718 u^{2}

数字相乘:

4718 \cdot 1 = 4718

= - 4718 - 4718 u^{2}

= 6700 + 13750 u - 4718 - 4718 u^{2}

化简

6700 + 13750 u - 4718 - 4718 u^{2} : - 4718 u^{2} + 13750 u + 1982

6700 + 13750 u - 4718 - 4718 u^{2}

对同类项分组

= - 4718 u^{2} + 13750 u + 6700 - 4718

数字相加/相减:

6700 - 4718 = 1982

= - 4718 u^{2} + 13750 u + 1982

= - 4718 u^{2} + 13750 u + 1982

- 4718 u^{2} + 13750 u + 1982 = 10000

将

10000

para o lado esquerdo

- 4718 u^{2} + 13750 u + 1982 = 10000

两边减去

10000

- 4718 u^{2} + 13750 u + 1982 - 10000 = 10000 - 10000

化简

- 4718 u^{2} + 13750 u - 8018 = 0

- 4718 u^{2} + 13750 u - 8018 = 0

使用求根公式求解

- 4718 u^{2} + 13750 u - 8018 = 0

二次方程求根公式:

若

a = - 4718, b = 13750, c = - 8018

u_{1, 2} = \frac{- 13750 \pm 1375 0 ^{2} - 4 ( - 4718 ) ( - 8018 )}{2 ( - 4718 )}

u_{1, 2} = \frac{- 13750 \pm 1375 0 ^{2} - 4 ( - 4718 ) ( - 8018 )}{2 ( - 4718 )}

1375 0^{2} - 4 (- 4718) (- 8018) = 29436701

1375 0^{2} - 4 (- 4718) (- 8018)

使用法则

- (- a) = a

= 1375 0^{2} - 4 \cdot 4718 \cdot 8018

数字相乘:

4 \cdot 4718 \cdot 8018 = 151315696

= 1375 0^{2} - 151315696

1375 0^{2} = 189062500

= 189062500 - 151315696

数字相减:

189062500 - 151315696 = 37746804

= 37746804

37746804

质因数分解:

2^{2} \cdot 3 \cdot 179 \cdot 17573

37746804

= 2^{2} \cdot 3 \cdot 179 \cdot 17573

使用根式运算法则:

n ab = n a n b

= 2^{2} 3 \cdot 179 \cdot 17573

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2 3 \cdot 179 \cdot 17573

整理后得

= 29436701

u_{1, 2} = \frac{- 13750 \pm 2 9436701}{2 ( - 4718 )}

将解分隔开

u_{1} = \frac{- 13750 + 2 9436701}{2 ( - 4718 )}, u_{2} = \frac{- 13750 - 2 9436701}{2 ( - 4718 )}

u = \frac{- 13750 + 2 9436701}{2 ( - 4718 )} : \frac{6875 - 9436701}{4718}

\frac{- 13750 + 2 9436701}{2 ( - 4718 )}

去除括号:

(- a) = - a

= \frac{- 13750 + 2 9436701}{- 2 \cdot 4718}

数字相乘:

2 \cdot 4718 = 9436

= \frac{- 13750 + 2 9436701}{- 9436}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

- 13750 + 29436701 = - (13750 - 29436701)

= \frac{13750 - 2 9436701}{9436}

分解

13750 - 29436701 : 2 (6875 - 9436701)

13750 - 29436701

改写为

= 2 \cdot 6875 - 29436701

因式分解出通项

2

= 2 (6875 - 9436701)

= \frac{2 ( 6875 - 9436701 )}{9436}

约分:

2

= \frac{6875 - 9436701}{4718}

u = \frac{- 13750 - 2 9436701}{2 ( - 4718 )} : \frac{6875 + 9436701}{4718}

\frac{- 13750 - 2 9436701}{2 ( - 4718 )}

去除括号:

(- a) = - a

= \frac{- 13750 - 2 9436701}{- 2 \cdot 4718}

数字相乘:

2 \cdot 4718 = 9436

= \frac{- 13750 - 2 9436701}{- 9436}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

- 13750 - 29436701 = - (13750 + 29436701)

= \frac{13750 + 2 9436701}{9436}

分解

13750 + 29436701 : 2 (6875 + 9436701)

13750 + 29436701

改写为

= 2 \cdot 6875 + 29436701

因式分解出通项

2

= 2 (6875 + 9436701)

= \frac{2 ( 6875 + 9436701 )}{9436}

约分:

2

= \frac{6875 + 9436701}{4718}

二次方程组的解是:

u = \frac{6875 - 9436701}{4718}, u = \frac{6875 + 9436701}{4718}

u = tan (θ)

代回

tan (θ) = \frac{6875 - 9436701}{4718}, tan (θ) = \frac{6875 + 9436701}{4718}

tan (θ) = \frac{6875 - 9436701}{4718}, tan (θ) = \frac{6875 + 9436701}{4718}

tan (θ) = \frac{6875 - 9436701}{4718} : θ = arctan (\frac{6875 - 9436701}{4718}) + πn

tan (θ) = \frac{6875 - 9436701}{4718}

使用反三角函数性质

tan (θ) = \frac{6875 - 9436701}{4718}

tan (θ) = \frac{6875 - 9436701}{4718}

的通解

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

θ = arctan (\frac{6875 - 9436701}{4718}) + πn

θ = arctan (\frac{6875 - 9436701}{4718}) + πn

tan (θ) = \frac{6875 + 9436701}{4718} : θ = arctan (\frac{6875 + 9436701}{4718}) + πn

tan (θ) = \frac{6875 + 9436701}{4718}

使用反三角函数性质

tan (θ) = \frac{6875 + 9436701}{4718}

tan (θ) = \frac{6875 + 9436701}{4718}

的通解

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

θ = arctan (\frac{6875 + 9436701}{4718}) + πn

θ = arctan (\frac{6875 + 9436701}{4718}) + πn

合并所有解

θ = arctan (\frac{6875 - 9436701}{4718}) + πn, θ = arctan (\frac{6875 + 9436701}{4718}) + πn

以小数形式表示解

θ = 0.67843 \dots + πn, θ = 1.12790 \dots + πn

作图

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cos(θ)=-cos(θ)

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