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受欢迎的三角函数 >

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

解答

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

解答

x = 2 πn, x = π + 2 πn, x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = - 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 199.47122 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

3 cos^{2} (x)

3 + sin (x) - 3 cos^{2} (x) = 0

使用三角恒等式改写

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

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

化简

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = - 3, b = 1, c = sin^{2} (x)

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

使用加减运算法则

- (- a) = a

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

数字相乘:

3 \cdot 1 = 3

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

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

化简

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

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

对同类项分组

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

3 - 3 = 0

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

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

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

sin (x) + 3 sin^{2} (x) = 0

用替代法求解

sin (x) + 3 sin^{2} (x) = 0

令

: sin (x) = u

u + 3 u^{2} = 0

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

u + 3 u^{2} = 0

改写成标准形式

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

3 u^{2} + u = 0

使用求根公式求解

3 u^{2} + u = 0

二次方程求根公式:

若

a = 3, b = 1, c = 0

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 3 \cdot 0}{2 \cdot 3}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 3 \cdot 0}{2 \cdot 3}

1^{2} - 4 \cdot 3 \cdot 0 = 1

1^{2} - 4 \cdot 3 \cdot 0

使用法则

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 3 \cdot 0

使用法则

0 \cdot a = 0

= 1 - 0

数字相减:

1 - 0 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- 1 \pm 1}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- 1 + 1}{2 \cdot 3}, u_{2} = \frac{- 1 - 1}{2 \cdot 3}

u = \frac{- 1 + 1}{2 \cdot 3} : 0

\frac{- 1 + 1}{2 \cdot 3}

数字相加/相减:

- 1 + 1 = 0

= \frac{0}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{0}{6}

使用法则

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

= 0

u = \frac{- 1 - 1}{2 \cdot 3} : - \frac{1}{3}

\frac{- 1 - 1}{2 \cdot 3}

数字相减:

- 1 - 1 = - 2

= \frac{- 2}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 2}{6}

使用分式法则:

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

= - \frac{2}{6}

约分:

2

= - \frac{1}{3}

二次方程组的解是:

u = 0, u = - \frac{1}{3}

u = sin (x)

代回

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

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

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{1}{3} : x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

sin (x) = - \frac{1}{3}

使用反三角函数性质

sin (x) = - \frac{1}{3}

sin (x) = - \frac{1}{3}

的通解

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

x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

以小数形式表示解

x = 2 πn, x = π + 2 πn, x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn

作图

流行的例子

cot (θ) = 3

sin^2(x)+4sin(x)=0

sin^{2} (x) + 4 sin (x) = 0

(tan(x)-1)(sec(x)-1)=0

(tan (x) - 1) (sec (x) - 1) = 0

4 - 8 sin^{2} (x) = 0

1-sin(θ)=sqrt(3)cos(θ)

1 - sin (θ) = 3 cos (θ)