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受欢迎的三角函数 >

sqrt(5)tan(θ)-9=-6tan(θ)+sqrt(8)

解答

5 tan (θ) - 9 = - 6 tan (θ) + 8

解答

θ = 0.96256 \dots + πn

+1

度数

θ = 55.15072 \dots^{\circ} + 18 0^{\circ} n

求解步骤

5 tan (θ) - 9 = - 6 tan (θ) + 8

用替代法求解

5 tan (θ) - 9 = - 6 tan (θ) + 8

令

: tan (θ) = u

5 u - 9 = - 6 u + 8

5 u - 9 = - 6 u + 8 : u = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

5 u - 9 = - 6 u + 8

化简

- 6 u + 8 : - 6 u + 22

- 6 u + 8

8 = 22

8

8

质因数分解:

2^{3}

8

8

除以

28 = 4 \cdot 2

= 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

使用根式运算法则:

n ab = n a n b

= 2 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 22

= - 6 u + 22

5 u - 9 = - 6 u + 22

将

9

到右边

5 u - 9 = - 6 u + 22

两边加上

9

5 u - 9 + 9 = - 6 u + 22 + 9

化简

5 u = - 6 u + 22 + 9

5 u = - 6 u + 22 + 9

将

6 u

para o lado esquerdo

5 u = - 6 u + 22 + 9

两边加上

6 u

5 u + 6 u = - 6 u + 22 + 9 + 6 u

化简

5 u + 6 u = 22 + 9

5 u + 6 u = 22 + 9

分解

5 u + 6 u : (5 + 6) u

5 u + 6 u

因式分解出通项

u

= u (5 + 6)

(5 + 6) u = 22 + 9

两边除以

5 + 6

(5 + 6) u = 22 + 9

两边除以

5 + 6

\frac{( 5 + 6 ) u}{5 + 6} = \frac{2 2}{5 + 6} + \frac{9}{5 + 6}

化简

\frac{( 5 + 6 ) u}{5 + 6} = \frac{2 2}{5 + 6} + \frac{9}{5 + 6}

化简

\frac{( 5 + 6 ) u}{5 + 6} : u

\frac{( 5 + 6 ) u}{5 + 6}

约分:

5 + 6

= u

化简

\frac{2 2}{5 + 6} + \frac{9}{5 + 6} : - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

\frac{2 2}{5 + 6} + \frac{9}{5 + 6}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 2 + 9}{5 + 6}

乘以共轭根式

\frac{5 - 6}{5 - 6}

= \frac{( 2 2 + 9 ) ( 5 - 6 )}{( 5 + 6 ) ( 5 - 6 )}

(5 + 6) (5 - 6) = - 31

(5 + 6) (5 - 6)

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

a = 5, b = 6

= (5)^{2} - 6^{2}

化简

(5)^{2} - 6^{2} : - 31

(5)^{2} - 6^{2}

(5)^{2} = 5

(5)^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

= (5^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 5^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 5

6^{2} = 36

6^{2}

6^{2} = 36

= 36

= 5 - 36

数字相减:

5 - 36 = - 31

= - 31

= - 31

= \frac{( 2 2 + 9 ) ( 5 - 6 )}{- 31}

使用分式法则:

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

= - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

u = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

u = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

u = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

u = tan (θ)

代回

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31} : θ = arctan (- \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}) + πn

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

使用反三角函数性质

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

tan (θ) = - \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}

的通解

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

θ = arctan (- \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}) + πn

θ = arctan (- \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}) + πn

合并所有解

θ = arctan (- \frac{( 2 2 + 9 ) ( 5 - 6 )}{31}) + πn

以小数形式表示解

θ = 0.96256 \dots + πn

作图

流行的例子

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6 cos^{2} (x) + sin (x) - 5 = 0

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sin (θ - \frac{π}{2}) = cos (θ)