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受欢迎的三角函数 >

2sin^2(x)+(-2-sqrt(2))sin(x)+sqrt(2)=0

解答

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

解答

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

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: sin (x) = u

2 u^{2} + (- 2 - 2) u + 2 = 0

2 u^{2} + (- 2 - 2) u + 2 = 0 : u = 1, u = \frac{2}{2}

2 u^{2} + (- 2 - 2) u + 2 = 0

展开

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

2 u^{2} + (- 2 - 2) u + 2

乘开

u (- 2 - 2) : - 2 u - 2 u

u (- 2 - 2)

使用分配律:

a (b - c) = ab - a c

a = u, b = - 2, c = 2

= u (- 2) - u 2

使用加减运算法则

+ (- a) = - a

= - 2 u - 2 u

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

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

改写成标准形式

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

2 u^{2} - (2 + 2) u + 2 = 0

使用求根公式求解

2 u^{2} - (2 + 2) u + 2 = 0

二次方程求根公式:

若

a = 2, b = - 2 - 2, c = 2

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

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

(- 2 - 2)^{2} - 4 \cdot 22 = 2 - 2

(- 2 - 2)^{2} - 4 \cdot 22

数字相乘:

4 \cdot 2 = 8

= (- 2 - 2)^{2} - 82

乘开

(- 2 - 2)^{2} - 82 : 6 - 42

(- 2 - 2)^{2} - 82

(- 2 - 2)^{2} : 6 + 42

使用完全平方公式:

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

a = - 2, b = 2

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

化简

(- 2)^{2} - 2 (- 2) 2 + (2)^{2} : 6 + 42

(- 2)^{2} - 2 (- 2) 2 + (2)^{2}

使用法则

- (- a) = a

= (- 2)^{2} + 2 \cdot 22 + (2)^{2}

(- 2)^{2} = 4

(- 2)^{2}

使用指数法则:

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

若

n

是偶数

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

= 2^{2}

2^{2} = 4

= 4

2 \cdot 22 = 42

2 \cdot 22

数字相乘:

2 \cdot 2 = 4

= 42

(2)^{2} = 2

(2)^{2}

使用根式运算法则:

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

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

使用指数法则:

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

= 2^{\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

= 2

= 4 + 42 + 2

数字相加:

4 + 2 = 6

= 6 + 42

= 6 + 42

= 6 + 42 - 82

同类项相加:

42 - 82 = - 42

= 6 - 42

= 6 - 42

= 2 - 42 + 4

= (2)^{2} - 42 + (4)^{2}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= (2)^{2} - 42 + 2^{2}

22 \cdot 2 = 42

22 \cdot 2

数字相乘:

2 \cdot 2 = 4

= 42

= (2)^{2} - 22 \cdot 2 + 2^{2}

使用完全平方公式:

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

(2)^{2} - 22 \cdot 2 + 2^{2} = (2 - 2)^{2}

= (2 - 2)^{2}

使用指数法则:

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

若

n

是偶数

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

= (2 - 2)^{2}

使用根式运算法则:

n a^{n} = a

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

= 2 - 2

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

将解分隔开

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

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

\frac{- ( - 2 - 2 ) + 2 - 2}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- ( - 2 - 2 ) + 2 - 2}{4}

乘开

- (- 2 - 2) + 2 - 2 : 4

- (- 2 - 2) + 2 - 2

- (- 2 - 2) : 2 + 2

- (- 2 - 2)

打开括号

= - (- 2) - (- 2)

使用加减运算法则

- (- a) = a

= 2 + 2

= 2 + 2 + 2 - 2

化简

2 + 2 + 2 - 2 : 4

2 + 2 + 2 - 2

同类项相加:

2 - 2 = 0

= 2 + 2

数字相加:

2 + 2 = 4

= 4

= 4

= \frac{4}{4}

使用法则

\frac{a}{a} = 1

= 1

u = \frac{- ( - 2 - 2 ) - ( 2 - 2 )}{2 \cdot 2} : \frac{2}{2}

\frac{- ( - 2 - 2 ) - ( 2 - 2 )}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- ( - 2 - 2 ) - ( 2 - 2 )}{4}

乘开

- (- 2 - 2) - (2 - 2) : 22

- (- 2 - 2) - (2 - 2)

- (- 2 - 2) : 2 + 2

- (- 2 - 2)

打开括号

= - (- 2) - (- 2)

使用加减运算法则

- (- a) = a

= 2 + 2

= 2 + 2 - (2 - 2)

- (2 - 2) : - 2 + 2

- (2 - 2)

打开括号

= - (2) - (- 2)

使用加减运算法则

- (- a) = a, - (a) = - a

= - 2 + 2

= 2 + 2 - 2 + 2

化简

2 + 2 - 2 + 2 : 22

2 + 2 - 2 + 2

同类项相加:

2 + 2 = 22

= 2 + 22 - 2

2 - 2 = 0

= 22

= 22

= \frac{2 2}{4}

约分:

2

= \frac{2}{2}

二次方程组的解是:

u = 1, u = \frac{2}{2}

u = sin (x)

代回

sin (x) = 1, sin (x) = \frac{2}{2}

sin (x) = 1, sin (x) = \frac{2}{2}

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

sin (x) = 1

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 = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

sin (x) = \frac{2}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = \frac{2}{2}

sin (x) = \frac{2}{2}

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

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

合并所有解

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

作图

流行的例子

3-sqrt(3)tan(2x)=4-2sqrt(3)tan(2x)

3 - 3 tan (2 x) = 4 - 23 tan (2 x)

cos(θ)= 3/(sqrt(34))

cos (θ) = \frac{3}{34}

tan(θ)+cot(θ)= 4/(sqrt(3))

tan (θ) + cot (θ) = \frac{4}{3}

sec(x)+4cos(x)=5

sec (x) + 4 cos (x) = 5

-sqrt(3)sec(x)=2

- 3 sec (x) = 2