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受欢迎的三角函数 >

cos^2(x)=sin^2(x)-(sqrt(2))/2

解答

cos^{2} (x) = sin^{2} (x) - \frac{2}{2}

解答

x = 1.17809 \dots + 2 πn, x = π - 1.17809 \dots + 2 πn, x = - 1.17809 \dots + 2 πn, x = π + 1.17809 \dots + 2 πn

+1

度数

x = 67. 5^{\circ} + 36 0^{\circ} n, x = 112. 5^{\circ} + 36 0^{\circ} n, x = - 67. 5^{\circ} + 36 0^{\circ} n, x = 247. 5^{\circ} + 36 0^{\circ} n

求解步骤

cos^{2} (x) = sin^{2} (x) - \frac{2}{2}

两边减去

sin^{2} (x) - \frac{2}{2}

cos^{2} (x) - sin^{2} (x) + \frac{1}{2} = 0

化简

cos^{2} (x) - sin^{2} (x) + \frac{1}{2} : \frac{2 cos ^{2} ( x ) - 2 sin ^{2} ( x ) + 1}{2}

cos^{2} (x) - sin^{2} (x) + \frac{1}{2}

将项转换为分式:

cos^{2} (x) = \frac{cos ^{2} ( x ) 2}{2}, sin^{2} (x) = \frac{sin ^{2} ( x ) 2}{2}

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

因为分母相等，所以合并分式:

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

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

\frac{2 cos ^{2} ( x ) - 2 sin ^{2} ( x ) + 1}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

化简

1 + (1 - sin^{2} (x)) 2 - sin^{2} (x) 2 : 1 + 2 - 22 sin^{2} (x)

1 + (1 - sin^{2} (x)) 2 - sin^{2} (x) 2

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

乘开

2 (1 - sin^{2} (x)) : 2 - 2 sin^{2} (x)

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

使用分配律:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (x)

= 2 \cdot 1 - 2 sin^{2} (x)

= 1 \cdot 2 - 2 sin^{2} (x)

乘以:

1 \cdot 2 = 2

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

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

同类项相加:

- 2 sin^{2} (x) - 2 sin^{2} (x) = - 22 sin^{2} (x)

= 1 + 2 - 22 sin^{2} (x)

= 1 + 2 - 22 sin^{2} (x)

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

用替代法求解

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

令

: sin (x) = u

1 + 2 - 2 u^{2} 2 = 0

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

1 + 2 - 2 u^{2} 2 = 0

将

1

到右边

1 + 2 - 2 u^{2} 2 = 0

两边减去

1

1 + 2 - 2 u^{2} 2 - 1 = 0 - 1

化简

2 - 2 u^{2} 2 = - 1

2 - 2 u^{2} 2 = - 1

将

2

到右边

2 - 2 u^{2} 2 = - 1

两边减去

2

2 - 2 u^{2} 2 - 2 = - 1 - 2

化简

- 2 u^{2} 2 = - 1 - 2

- 2 u^{2} 2 = - 1 - 2

两边除以

- 22

- 2 u^{2} 2 = - 1 - 2

两边除以

- 22

\frac{- 2 u ^{2} 2}{- 2 2} = - \frac{1}{- 2 2} - \frac{2}{- 2 2}

化简

\frac{- 2 u ^{2} 2}{- 2 2} = - \frac{1}{- 2 2} - \frac{2}{- 2 2}

化简

\frac{- 2 u ^{2} 2}{- 2 2} : u^{2}

\frac{- 2 u ^{2} 2}{- 2 2}

使用分式法则:

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

= \frac{2 u ^{2} 2}{2 2}

数字相除:

\frac{2}{2} = 1

= \frac{2 u ^{2}}{2}

约分:

2

= u^{2}

化简

- \frac{1}{- 2 2} - \frac{2}{- 2 2} : \frac{2 + 2}{4}

- \frac{1}{- 2 2} - \frac{2}{- 2 2}

使用法则

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

= \frac{- 1 - 2}{- 2 2}

使用分式法则:

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

- 1 - 2 = - (1 + 2)

= \frac{1 + 2}{2 2}

\frac{1 + 2}{2 2}

有理化:

\frac{2 + 2}{2 ^{2}}

\frac{1 + 2}{2 2}

乘以共轭根式

\frac{2}{2}

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

(1 + 2) 2 = 2 + 2

(1 + 2) 2

= 2 (1 + 2)

使用分配律:

a (b + c) = ab + a c

a = 2, b = 1, c = 2

= 2 \cdot 1 + 22

= 1 \cdot 2 + 22

化简

1 \cdot 2 + 22 : 2 + 2

1 \cdot 2 + 22

乘以:

1 \cdot 2 = 2

= 2 + 22

使用根式运算法则:

a a = a

22 = 2

= 2 + 2

= 2 + 2

222 = 2^{2}

222

使用指数法则:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

同类项相加:

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

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

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

2 \cdot \frac{1}{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^{1 + 1}

数字相加:

1 + 1 = 2

= 2^{2}

= \frac{2 + 2}{2 ^{2}}

= \frac{2 + 2}{2 ^{2}}

2^{2} = 4

= \frac{2 + 2}{4}

u^{2} = \frac{2 + 2}{4}

u^{2} = \frac{2 + 2}{4}

u^{2} = \frac{2 + 2}{4}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{2 + 2}{4}, u = - \frac{2 + 2}{4}

\frac{2 + 2}{4} = \frac{2 + 2}{2}

\frac{2 + 2}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{2 + 2}{4}

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

= \frac{2 + 2}{2}

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

- \frac{2 + 2}{4}

化简

\frac{2 + 2}{4} : \frac{2 + 2}{2}

\frac{2 + 2}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{2 + 2}{4}

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

= \frac{2 + 2}{2}

= - \frac{2 + 2}{2}

= - \frac{2 + 2}{2}

u = \frac{2 + 2}{2}, u = - \frac{2 + 2}{2}

u = sin (x)

代回

sin (x) = \frac{2 + 2}{2}, sin (x) = - \frac{2 + 2}{2}

sin (x) = \frac{2 + 2}{2}, sin (x) = - \frac{2 + 2}{2}

sin (x) = \frac{2 + 2}{2} : x = arcsin (\frac{2 + 2}{2}) + 2 πn, x = π - arcsin (\frac{2 + 2}{2}) + 2 πn

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

使用反三角函数性质

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

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

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{2 + 2}{2}) + 2 πn, x = π - arcsin (\frac{2 + 2}{2}) + 2 πn

x = arcsin (\frac{2 + 2}{2}) + 2 πn, x = π - arcsin (\frac{2 + 2}{2}) + 2 πn

sin (x) = - \frac{2 + 2}{2} : x = arcsin (- \frac{2 + 2}{2}) + 2 πn, x = π + arcsin (\frac{2 + 2}{2}) + 2 πn

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

使用反三角函数性质

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

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

的通解

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{2 + 2}{2}) + 2 πn, x = π + arcsin (\frac{2 + 2}{2}) + 2 πn

x = arcsin (- \frac{2 + 2}{2}) + 2 πn, x = π + arcsin (\frac{2 + 2}{2}) + 2 πn

合并所有解

x = arcsin (\frac{2 + 2}{2}) + 2 πn, x = π - arcsin (\frac{2 + 2}{2}) + 2 πn, x = arcsin (- \frac{2 + 2}{2}) + 2 πn, x = π + arcsin (\frac{2 + 2}{2}) + 2 πn

以小数形式表示解

x = 1.17809 \dots + 2 πn, x = π - 1.17809 \dots + 2 πn, x = - 1.17809 \dots + 2 πn, x = π + 1.17809 \dots + 2 πn

作图

流行的例子

cos(2θ)-sin(θ)=0

cos (2 θ) - sin (θ) = 0

tan(θ)=-(2sqrt(3))/3 sin(θ)

tan (θ) = - \frac{2 3}{3} sin (θ)

sqrt(3)cos(x)tan(x)-cos(x)=0

3 cos (x) tan (x) - cos (x) = 0

4 sin^{2} (θ) = 3

cos(2x)=sin(x)+1

cos (2 x) = sin (x) + 1