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受欢迎的三角函数 >

3cos^2(x)-sin(x)+1=0

解答

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

解答

x = \frac{π}{2} + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

1 - 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)

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

化简

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

1 - 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)

数字相乘:

3 \cdot 1 = 3

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

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

化简

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

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

对同类项分组

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

数字相加:

1 + 3 = 4

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

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

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

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

用替代法求解

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

令

: sin (x) = u

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

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 3, b = - 1, c = 4

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

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

(- 1)^{2} - 4 (- 3) \cdot 4 = 7

(- 1)^{2} - 4 (- 3) \cdot 4

使用法则

- (- a) = a

= (- 1)^{2} + 4 \cdot 3 \cdot 4

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

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

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 3 \cdot 4 = 48

4 \cdot 3 \cdot 4

数字相乘:

4 \cdot 3 \cdot 4 = 48

= 48

= 1 + 48

数字相加:

1 + 48 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 1 ) \pm 7}{2 ( - 3 )}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 7}{2 ( - 3 )}, u_{2} = \frac{- ( - 1 ) - 7}{2 ( - 3 )}

u = \frac{- ( - 1 ) + 7}{2 ( - 3 )} : - \frac{4}{3}

\frac{- ( - 1 ) + 7}{2 ( - 3 )}

去除括号:

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

= \frac{1 + 7}{- 2 \cdot 3}

数字相加:

1 + 7 = 8

= \frac{8}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{8}{- 6}

使用分式法则:

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

= - \frac{8}{6}

约分:

2

= - \frac{4}{3}

u = \frac{- ( - 1 ) - 7}{2 ( - 3 )} : 1

\frac{- ( - 1 ) - 7}{2 ( - 3 )}

去除括号:

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

= \frac{1 - 7}{- 2 \cdot 3}

数字相减:

1 - 7 = - 6

= \frac{- 6}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 6}{- 6}

使用分式法则:

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

= \frac{6}{6}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

u = - \frac{4}{3}, u = 1

u = sin (x)

代回

sin (x) = - \frac{4}{3}, sin (x) = 1

sin (x) = - \frac{4}{3}, sin (x) = 1

sin (x) = - \frac{4}{3} :

无解

sin (x) = - \frac{4}{3}

- 1 \leq sin (x) \leq 1

无解

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

合并所有解

x = \frac{π}{2} + 2 πn

作图

流行的例子

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

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

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

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

tan^{2} (b) = 3

solvefor a,sin(a+b/2)=cos(c/2)

so l v e f or a, sin (a + \frac{b}{2}) = cos (\frac{c}{2})

tan(x)-cot(x)=2cot^2(x)

tan (x) - cot (x) = 2 cot^{2} (x)