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受欢迎的三角函数 >

4sin^2(θ)-4cos(θ)=1

解答

4 sin^{2} (θ) - 4 cos (θ) = 1

解答

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

+1

度数

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

4 sin^{2} (θ) - 4 cos (θ) = 1

两边减去

1

4 sin^{2} (θ) - 4 cos (θ) - 1 = 0

使用三角恒等式改写

- 1 - 4 cos (θ) + 4 sin^{2} (θ)

使用毕达哥拉斯恒等式:

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

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

= - 1 - 4 cos (θ) + 4 (1 - cos^{2} (θ))

化简

- 1 - 4 cos (θ) + 4 (1 - cos^{2} (θ)) : - 4 cos^{2} (θ) - 4 cos (θ) + 3

- 1 - 4 cos (θ) + 4 (1 - cos^{2} (θ))

乘开

4 (1 - cos^{2} (θ)) : 4 - 4 cos^{2} (θ)

4 (1 - cos^{2} (θ))

使用分配律:

a (b - c) = ab - a c

a = 4, b = 1, c = cos^{2} (θ)

= 4 \cdot 1 - 4 cos^{2} (θ)

数字相乘:

4 \cdot 1 = 4

= 4 - 4 cos^{2} (θ)

= - 1 - 4 cos (θ) + 4 - 4 cos^{2} (θ)

化简

- 1 - 4 cos (θ) + 4 - 4 cos^{2} (θ) : - 4 cos^{2} (θ) - 4 cos (θ) + 3

- 1 - 4 cos (θ) + 4 - 4 cos^{2} (θ)

对同类项分组

= - 4 cos (θ) - 4 cos^{2} (θ) - 1 + 4

数字相加/相减:

- 1 + 4 = 3

= - 4 cos^{2} (θ) - 4 cos (θ) + 3

= - 4 cos^{2} (θ) - 4 cos (θ) + 3

= - 4 cos^{2} (θ) - 4 cos (θ) + 3

3 - 4 cos (θ) - 4 cos^{2} (θ) = 0

用替代法求解

3 - 4 cos (θ) - 4 cos^{2} (θ) = 0

令

: cos (θ) = u

3 - 4 u - 4 u^{2} = 0

3 - 4 u - 4 u^{2} = 0 : u = - \frac{3}{2}, u = \frac{1}{2}

3 - 4 u - 4 u^{2} = 0

改写成标准形式

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

- 4 u^{2} - 4 u + 3 = 0

使用求根公式求解

- 4 u^{2} - 4 u + 3 = 0

二次方程求根公式:

若

a = - 4, b = - 4, c = 3

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 ( - 4 ) \cdot 3}{2 ( - 4 )}

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 ( - 4 ) \cdot 3}{2 ( - 4 )}

(- 4)^{2} - 4 (- 4) \cdot 3 = 8

(- 4)^{2} - 4 (- 4) \cdot 3

使用法则

- (- a) = a

= (- 4)^{2} + 4 \cdot 4 \cdot 3

使用指数法则:

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

若

n

是偶数

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

= 4^{2} + 4 \cdot 4 \cdot 3

数字相乘:

4 \cdot 4 \cdot 3 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

数字相加:

16 + 48 = 64

= 64

因式分解数字:

64 = 8^{2}

= 8^{2}

使用根式运算法则:

n a^{n} = a

8^{2} = 8

= 8

u_{1, 2} = \frac{- ( - 4 ) \pm 8}{2 ( - 4 )}

将解分隔开

u_{1} = \frac{- ( - 4 ) + 8}{2 ( - 4 )}, u_{2} = \frac{- ( - 4 ) - 8}{2 ( - 4 )}

u = \frac{- ( - 4 ) + 8}{2 ( - 4 )} : - \frac{3}{2}

\frac{- ( - 4 ) + 8}{2 ( - 4 )}

去除括号:

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

= \frac{4 + 8}{- 2 \cdot 4}

数字相加:

4 + 8 = 12

= \frac{12}{- 2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{12}{- 8}

使用分式法则:

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

= - \frac{12}{8}

约分:

4

= - \frac{3}{2}

u = \frac{- ( - 4 ) - 8}{2 ( - 4 )} : \frac{1}{2}

\frac{- ( - 4 ) - 8}{2 ( - 4 )}

去除括号:

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

= \frac{4 - 8}{- 2 \cdot 4}

数字相减:

4 - 8 = - 4

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

数字相乘:

2 \cdot 4 = 8

= \frac{- 4}{- 8}

使用分式法则:

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

= \frac{4}{8}

约分:

4

= \frac{1}{2}

二次方程组的解是:

u = - \frac{3}{2}, u = \frac{1}{2}

u = cos (θ)

代回

cos (θ) = - \frac{3}{2}, cos (θ) = \frac{1}{2}

cos (θ) = - \frac{3}{2}, cos (θ) = \frac{1}{2}

cos (θ) = - \frac{3}{2} :

无解

cos (θ) = - \frac{3}{2}

- 1 \leq cos (x) \leq 1

无解

cos (θ) = \frac{1}{2} : θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

的通解

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

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

合并所有解

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

作图

流行的例子

(33)/(sin(x))=(29)/(sin(100))

\frac{33}{sin ( x )} = \frac{29}{sin ( 10 0 ^{\circ} )}

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

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

(\frac{1}{2}) = cos (θ)

sin(θ)= 4/5 ,θ,I,cos(2θ)

sin (θ) = \frac{4}{5}, θ, I, cos (2 θ)

sec^2(x)cot(x)=4

sec^{2} (x) cot (x) = 4