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受欢迎的三角函数 >

sin(x)=2*sin(3x)

解答

sin (x) = 2 \cdot sin (3 x)

解答

x = 2 πn, x = π + 2 πn, x = - 0.91173 \dots + 2 πn, x = π + 0.91173 \dots + 2 πn, x = 0.91173 \dots + 2 πn, x = π - 0.91173 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = - 52.23875 \dots^{\circ} + 36 0^{\circ} n, x = 232.23875 \dots^{\circ} + 36 0^{\circ} n, x = 52.23875 \dots^{\circ} + 36 0^{\circ} n, x = 127.76124 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin (x) = 2 sin (3 x)

两边减去

2 sin (3 x)

sin (x) - 2 sin (3 x) = 0

使用三角恒等式改写

sin (x) - 2 sin (3 x)

sin (3 x) = 3 sin (x) - 4 sin^{3} (x)

sin (3 x)

使用三角恒等式改写

sin (3 x)

改写为

= sin (2 x + x)

使用角和恒等式:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

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

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

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

化简

cos (2 x) sin (x) + cos (x) \cdot 2 sin (x) cos (x) : sin (x) cos (2 x) + 2 cos^{2} (x) sin (x)

cos (2 x) sin (x) + cos (x) 2 sin (x) cos (x)

cos (x) \cdot 2 sin (x) cos (x) = 2 cos^{2} (x) sin (x)

cos (x) 2 sin (x) cos (x)

使用指数法则:

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

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

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

数字相加:

1 + 1 = 2

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

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

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

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

使用倍角公式:

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

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

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

乘开

(1 - 2 sin^{2} (x)) sin (x) + 2 (1 - sin^{2} (x)) sin (x) : - 4 sin^{3} (x) + 3 sin (x)

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

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

乘开

sin (x) (1 - 2 sin^{2} (x)) : sin (x) - 2 sin^{3} (x)

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

使用分配律:

a (b - c) = ab - a c

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

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

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

化简

1 \cdot sin (x) - 2 sin^{2} (x) sin (x) : sin (x) - 2 sin^{3} (x)

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

1 \cdot sin (x) = sin (x)

1 sin (x)

乘以:

1 \cdot sin (x) = sin (x)

= sin (x)

2 sin^{2} (x) sin (x) = 2 sin^{3} (x)

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

使用指数法则:

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

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

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

数字相加:

2 + 1 = 3

= 2 sin^{3} (x)

= sin (x) - 2 sin^{3} (x)

= sin (x) - 2 sin^{3} (x)

= sin (x) - 2 sin^{3} (x) + 2 (1 - sin^{2} (x)) sin (x)

乘开

2 sin (x) (1 - sin^{2} (x)) : 2 sin (x) - 2 sin^{3} (x)

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

使用分配律:

a (b - c) = ab - a c

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

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

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

化简

2 \cdot 1 \cdot sin (x) - 2 sin^{2} (x) sin (x) : 2 sin (x) - 2 sin^{3} (x)

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

2 \cdot 1 \cdot sin (x) = 2 sin (x)

2 \cdot 1 sin (x)

数字相乘:

2 \cdot 1 = 2

= 2 sin (x)

2 sin^{2} (x) sin (x) = 2 sin^{3} (x)

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

使用指数法则:

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

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

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

数字相加:

2 + 1 = 3

= 2 sin^{3} (x)

= 2 sin (x) - 2 sin^{3} (x)

= 2 sin (x) - 2 sin^{3} (x)

= sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x)

化简

sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x) : - 4 sin^{3} (x) + 3 sin (x)

sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x)

对同类项分组

= - 2 sin^{3} (x) - 2 sin^{3} (x) + sin (x) + 2 sin (x)

同类项相加:

- 2 sin^{3} (x) - 2 sin^{3} (x) = - 4 sin^{3} (x)

= - 4 sin^{3} (x) + sin (x) + 2 sin (x)

同类项相加:

sin (x) + 2 sin (x) = 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= sin (x) - 2 (3 sin (x) - 4 sin^{3} (x))

化简

sin (x) - 2 (3 sin (x) - 4 sin^{3} (x)) : - 5 sin (x) + 8 sin^{3} (x)

sin (x) - 2 (3 sin (x) - 4 sin^{3} (x))

乘开

- 2 (3 sin (x) - 4 sin^{3} (x)) : - 6 sin (x) + 8 sin^{3} (x)

- 2 (3 sin (x) - 4 sin^{3} (x))

使用分配律:

a (b - c) = ab - a c

a = - 2, b = 3 sin (x), c = 4 sin^{3} (x)

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

使用加减运算法则

- (- a) = a

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

化简

- 2 \cdot 3 sin (x) + 2 \cdot 4 sin^{3} (x) : - 6 sin (x) + 8 sin^{3} (x)

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

数字相乘:

2 \cdot 3 = 6

= - 6 sin (x) + 2 \cdot 4 sin^{3} (x)

数字相乘:

2 \cdot 4 = 8

= - 6 sin (x) + 8 sin^{3} (x)

= - 6 sin (x) + 8 sin^{3} (x)

= sin (x) - 6 sin (x) + 8 sin^{3} (x)

同类项相加:

sin (x) - 6 sin (x) = - 5 sin (x)

= - 5 sin (x) + 8 sin^{3} (x)

= - 5 sin (x) + 8 sin^{3} (x)

- 5 sin (x) + 8 sin^{3} (x) = 0

用替代法求解

- 5 sin (x) + 8 sin^{3} (x) = 0

令

: sin (x) = u

- 5 u + 8 u^{3} = 0

- 5 u + 8 u^{3} = 0 : u = 0, u = - \frac{10}{4}, u = \frac{10}{4}

- 5 u + 8 u^{3} = 0

因式分解

- 5 u + 8 u^{3} : u (22 u + 5) (22 u - 5)

- 5 u + 8 u^{3}

因式分解出通项

u : u (8 u^{2} - 5)

8 u^{3} - 5 u

使用指数法则:

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

u^{3} = u^{2} u

= 8 u^{2} u - 5 u

因式分解出通项

u

= u (8 u^{2} - 5)

= u (8 u^{2} - 5)

分解

8 u^{2} - 5 : (8 u + 5) (8 u - 5)

8 u^{2} - 5

将

8 u^{2} - 5

改写为

(8 u)^{2} - (5)^{2}

8 u^{2} - 5

使用根式运算法则:

a = (a)^{2}

8 = (8)^{2}

= (8)^{2} u^{2} - 5

使用根式运算法则:

a = (a)^{2}

5 = (5)^{2}

= (8)^{2} u^{2} - (5)^{2}

使用指数法则:

a^{m} b^{m} = (ab)^{m}

(8)^{2} u^{2} = (8 u)^{2}

= (8 u)^{2} - (5)^{2}

= (8 u)^{2} - (5)^{2}

使用平方差公式:

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

(8 u)^{2} - (5)^{2} = (8 u + 5) (8 u - 5)

= (8 u + 5) (8 u - 5)

= u (8 u + 5) (8 u - 5)

整理后得

= u (22 u + 5) (22 u - 5)

u (22 u + 5) (22 u - 5) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or 22 u + 5 = 0 or 22 u - 5 = 0

解

22 u + 5 = 0 : u = - \frac{10}{4}

22 u + 5 = 0

将

5

到右边

22 u + 5 = 0

两边减去

5

22 u + 5 - 5 = 0 - 5

化简

22 u = - 5

22 u = - 5

两边除以

22

22 u = - 5

两边除以

22

\frac{2 2 u}{2 2} = \frac{- 5}{2 2}

化简

\frac{2 2 u}{2 2} = \frac{- 5}{2 2}

化简

\frac{2 2 u}{2 2} : u

\frac{2 2 u}{2 2}

数字相除:

\frac{2}{2} = 1

= \frac{2 u}{2}

约分:

2

= u

化简

\frac{- 5}{2 2} : - \frac{10}{4}

\frac{- 5}{2 2}

使用分式法则:

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

= - \frac{5}{2 2}

- \frac{5}{2 2}

有理化:

- \frac{10}{4}

- \frac{5}{2 2}

乘以共轭根式

\frac{2}{2}

= - \frac{5 2}{2 2 2}

52 = 10

52

使用根式运算法则:

a b = a \cdot b

52 = 5 \cdot 2

= 5 \cdot 2

数字相乘:

5 \cdot 2 = 10

= 10

222 = 4

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}

2^{2} = 4

= 4

= - \frac{10}{4}

= - \frac{10}{4}

u = - \frac{10}{4}

u = - \frac{10}{4}

u = - \frac{10}{4}

解

22 u - 5 = 0 : u = \frac{10}{4}

22 u - 5 = 0

将

5

到右边

22 u - 5 = 0

两边加上

5

22 u - 5 + 5 = 0 + 5

化简

22 u = 5

22 u = 5

两边除以

22

22 u = 5

两边除以

22

\frac{2 2 u}{2 2} = \frac{5}{2 2}

化简

\frac{2 2 u}{2 2} = \frac{5}{2 2}

化简

\frac{2 2 u}{2 2} : u

\frac{2 2 u}{2 2}

数字相除:

\frac{2}{2} = 1

= \frac{2 u}{2}

约分:

2

= u

化简

\frac{5}{2 2} : \frac{10}{4}

\frac{5}{2 2}

乘以共轭根式

\frac{2}{2}

= \frac{5 2}{2 2 2}

52 = 10

52

使用根式运算法则:

a b = a \cdot b

52 = 5 \cdot 2

= 5 \cdot 2

数字相乘:

5 \cdot 2 = 10

= 10

222 = 4

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}

2^{2} = 4

= 4

= \frac{10}{4}

u = \frac{10}{4}

u = \frac{10}{4}

u = \frac{10}{4}

解为

u = 0, u = - \frac{10}{4}, u = \frac{10}{4}

u = sin (x)

代回

sin (x) = 0, sin (x) = - \frac{10}{4}, sin (x) = \frac{10}{4}

sin (x) = 0, sin (x) = - \frac{10}{4}, sin (x) = \frac{10}{4}

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

sin (x) = - \frac{10}{4} : x = arcsin (- \frac{10}{4}) + 2 πn, x = π + arcsin (\frac{10}{4}) + 2 πn

sin (x) = - \frac{10}{4}

使用反三角函数性质

sin (x) = - \frac{10}{4}

sin (x) = - \frac{10}{4}

的通解

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

x = arcsin (- \frac{10}{4}) + 2 πn, x = π + arcsin (\frac{10}{4}) + 2 πn

x = arcsin (- \frac{10}{4}) + 2 πn, x = π + arcsin (\frac{10}{4}) + 2 πn

sin (x) = \frac{10}{4} : x = arcsin (\frac{10}{4}) + 2 πn, x = π - arcsin (\frac{10}{4}) + 2 πn

sin (x) = \frac{10}{4}

使用反三角函数性质

sin (x) = \frac{10}{4}

sin (x) = \frac{10}{4}

的通解

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

x = arcsin (\frac{10}{4}) + 2 πn, x = π - arcsin (\frac{10}{4}) + 2 πn

x = arcsin (\frac{10}{4}) + 2 πn, x = π - arcsin (\frac{10}{4}) + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = arcsin (- \frac{10}{4}) + 2 πn, x = π + arcsin (\frac{10}{4}) + 2 πn, x = arcsin (\frac{10}{4}) + 2 πn, x = π - arcsin (\frac{10}{4}) + 2 πn

以小数形式表示解

x = 2 πn, x = π + 2 πn, x = - 0.91173 \dots + 2 πn, x = π + 0.91173 \dots + 2 πn, x = 0.91173 \dots + 2 πn, x = π - 0.91173 \dots + 2 πn

作图

流行的例子

tan^2(x)-5tan(x)-6=0

tan^{2} (x) - 5 tan (x) - 6 = 0

tan (θ) = \frac{12}{5}

2cos^2(t)=1-cos(t)

2 cos^{2} (t) = 1 - cos (t)

sec^2(θ)+sec(θ)=0

sec^{2} (θ) + sec (θ) = 0

cos(2x)-3cos(x)+2=0

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