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受欢迎的三角函数 >

4sin^2(x)+cos(x)+1=0

解答

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

解答

x = π + 2 πn

+1

度数

x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

1 + cos (x) + 4 sin^{2} (x)

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 + cos (x) + 4 (1 - cos^{2} (x))

化简

1 + cos (x) + 4 (1 - cos^{2} (x)) : cos (x) - 4 cos^{2} (x) + 5

1 + cos (x) + 4 (1 - cos^{2} (x))

乘开

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

4 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 4, b = 1, c = cos^{2} (x)

= 4 \cdot 1 - 4 cos^{2} (x)

数字相乘:

4 \cdot 1 = 4

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

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

化简

1 + cos (x) + 4 - 4 cos^{2} (x) : cos (x) - 4 cos^{2} (x) + 5

1 + cos (x) + 4 - 4 cos^{2} (x)

对同类项分组

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

数字相加:

1 + 4 = 5

= cos (x) - 4 cos^{2} (x) + 5

= cos (x) - 4 cos^{2} (x) + 5

= cos (x) - 4 cos^{2} (x) + 5

5 + cos (x) - 4 cos^{2} (x) = 0

用替代法求解

5 + cos (x) - 4 cos^{2} (x) = 0

令

: cos (x) = u

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

5 + u - 4 u^{2} = 0 : u = - 1, u = \frac{5}{4}

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

改写成标准形式

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

- 4 u^{2} + u + 5 = 0

使用求根公式求解

- 4 u^{2} + u + 5 = 0

二次方程求根公式:

若

a = - 4, b = 1, c = 5

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 4 ) \cdot 5}{2 ( - 4 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 4 ) \cdot 5}{2 ( - 4 )}

1^{2} - 4 (- 4) \cdot 5 = 9

1^{2} - 4 (- 4) \cdot 5

使用法则

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 4) \cdot 5

使用法则

- (- a) = a

= 1 + 4 \cdot 4 \cdot 5

数字相乘:

4 \cdot 4 \cdot 5 = 80

= 1 + 80

数字相加:

1 + 80 = 81

= 81

因式分解数字:

81 = 9^{2}

= 9^{2}

使用根式运算法则:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- 1 \pm 9}{2 ( - 4 )}

将解分隔开

u_{1} = \frac{- 1 + 9}{2 ( - 4 )}, u_{2} = \frac{- 1 - 9}{2 ( - 4 )}

u = \frac{- 1 + 9}{2 ( - 4 )} : - 1

\frac{- 1 + 9}{2 ( - 4 )}

去除括号:

(- a) = - a

= \frac{- 1 + 9}{- 2 \cdot 4}

数字相加/相减:

- 1 + 9 = 8

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

数字相乘:

2 \cdot 4 = 8

= \frac{8}{- 8}

使用分式法则:

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

= - \frac{8}{8}

使用法则

\frac{a}{a} = 1

= - 1

u = \frac{- 1 - 9}{2 ( - 4 )} : \frac{5}{4}

\frac{- 1 - 9}{2 ( - 4 )}

去除括号:

(- a) = - a

= \frac{- 1 - 9}{- 2 \cdot 4}

数字相减:

- 1 - 9 = - 10

= \frac{- 10}{- 2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 10}{- 8}

使用分式法则:

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

= \frac{10}{8}

约分:

2

= \frac{5}{4}

二次方程组的解是:

u = - 1, u = \frac{5}{4}

u = cos (x)

代回

cos (x) = - 1, cos (x) = \frac{5}{4}

cos (x) = - 1, cos (x) = \frac{5}{4}

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

cos (x) = - 1

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}

x = π + 2 πn

x = π + 2 πn

cos (x) = \frac{5}{4} :

无解

cos (x) = \frac{5}{4}

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = π + 2 πn

作图

流行的例子

(cot^2(x))-csc(x)=1

(cot^{2} (x)) - csc (x) = 1

solvefor x,u=arctan(x/y)

so l v e f or x, u = arctan (\frac{x}{y})

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

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

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

sqrt(3)*sin(x)=cos(x)

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