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受欢迎的三角函数 >

sqrt(7)*sin^2(x)+cos^2(x)-1=6sin(x)

解答

7 \cdot sin^{2} (x) + cos^{2} (x) - 1 = 6 sin (x)

解答

x = 2 πn, x = π + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

7 sin^{2} (x) + cos^{2} (x) - 1 = 6 sin (x)

两边减去

6 sin (x)

7 sin^{2} (x) + cos^{2} (x) - 1 - 6 sin (x) = 0

使用三角恒等式改写

- 1 + cos^{2} (x) - 6 sin (x) + sin^{2} (x) 7

使用毕达哥拉斯恒等式:

1 = cos^{2} (x) + sin^{2} (x)

1 - cos^{2} (x) = sin^{2} (x)

= - 6 sin (x) + sin^{2} (x) 7 - sin^{2} (x)

- sin^{2} (x) - 6 sin (x) + sin^{2} (x) 7 = 0

用替代法求解

- sin^{2} (x) - 6 sin (x) + sin^{2} (x) 7 = 0

令

: sin (x) = u

- u^{2} - 6 u + u^{2} 7 = 0

- u^{2} - 6 u + u^{2} 7 = 0 : u = 1 + 7, u = 0

- u^{2} - 6 u + u^{2} 7 = 0

改写成标准形式

a x^{2} + b x + c = 0

(- 1 + 7) u^{2} - 6 u = 0

使用求根公式求解

(- 1 + 7) u^{2} - 6 u = 0

二次方程求根公式:

若

a = - 1 + 7, b = - 6, c = 0

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 ( - 1 + 7 ) \cdot 0}{2 ( - 1 + 7 )}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 ( - 1 + 7 ) \cdot 0}{2 ( - 1 + 7 )}

(- 6)^{2} - 4 (- 1 + 7) \cdot 0 = 6

(- 6)^{2} - 4 (- 1 + 7) \cdot 0

使用指数法则:

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

若

n

是偶数

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

= 6^{2} - 4 \cdot 0 \cdot (7 - 1)

使用法则

0 \cdot a = 0

= 6^{2} - 0

6^{2} - 0 = 6^{2}

= 6^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

= 6

u_{1, 2} = \frac{- ( - 6 ) \pm 6}{2 ( - 1 + 7 )}

将解分隔开

u_{1} = \frac{- ( - 6 ) + 6}{2 ( - 1 + 7 )}, u_{2} = \frac{- ( - 6 ) - 6}{2 ( - 1 + 7 )}

u = \frac{- ( - 6 ) + 6}{2 ( - 1 + 7 )} : 1 + 7

\frac{- ( - 6 ) + 6}{2 ( - 1 + 7 )}

使用法则

- (- a) = a

= \frac{6 + 6}{2 ( - 1 + 7 )}

数字相加:

6 + 6 = 12

= \frac{12}{2 ( 7 - 1 )}

数字相除:

\frac{12}{2} = 6

= \frac{6}{( - 1 + 7 )}

去除括号:

(- a) = - a

= \frac{6}{- 1 + 7}

\frac{6}{- 1 + 7}

有理化:

1 + 7

\frac{6}{- 1 + 7}

乘以共轭根式

\frac{1 + 7}{1 + 7}

= \frac{6 ( 1 + 7 )}{( - 1 + 7 ) ( 1 + 7 )}

(- 1 + 7) (1 + 7) = 6

(- 1 + 7) (1 + 7)

使用平方差公式:

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

a = 7, b = 1

= (7)^{2} - 1^{2}

化简

(7)^{2} - 1^{2} : 6

(7)^{2} - 1^{2}

使用法则

1^{a} = 1

1^{2} = 1

= (7)^{2} - 1

(7)^{2} = 7

(7)^{2}

使用根式运算法则:

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

= (7^{\frac{1}{2}})^{2}

使用指数法则:

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

= 7^{\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

= 7

= 7 - 1

数字相减:

7 - 1 = 6

= 6

= 6

= \frac{6 ( 1 + 7 )}{6}

数字相除:

\frac{6}{6} = 1

= 1 + 7

= 1 + 7

u = \frac{- ( - 6 ) - 6}{2 ( - 1 + 7 )} : 0

\frac{- ( - 6 ) - 6}{2 ( - 1 + 7 )}

使用法则

- (- a) = a

= \frac{6 - 6}{2 ( - 1 + 7 )}

数字相减:

6 - 6 = 0

= \frac{0}{2 ( 7 - 1 )}

使用法则

\frac{0}{a} = 0, a \neq = 0

= 0

二次方程组的解是:

u = 1 + 7, u = 0

u = sin (x)

代回

sin (x) = 1 + 7, sin (x) = 0

sin (x) = 1 + 7, sin (x) = 0

sin (x) = 1 + 7 :

无解

sin (x) = 1 + 7

- 1 \leq sin (x) \leq 1

无解

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn

作图

流行的例子

4sec^2(x)-3tan^2(x)=5tan^2(x)

4 sec^{2} (x) - 3 tan^{2} (x) = 5 tan^{2} (x)

cos^{3} (x) = 66

2sqrt(3)*sin(4x+60^0)-3=0

23 \cdot sin (4 x + 6 0^{0}) - 3 = 0

(sin(x)+sin^2(x))/2 =0.5

\frac{sin ( x ) + sin ^{2} ( x )}{2} = 0.5

cos (b) = \frac{3}{5}