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受欢迎的三角函数 >

2cos^2(x)+5sin(x)=5

解答

2 cos^{2} (x) + 5 sin (x) = 5

解答

x = \frac{π}{2} + 2 πn

+1

度数

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

求解步骤

2 cos^{2} (x) + 5 sin (x) = 5

两边减去

5

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

使用三角恒等式改写

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

= - 5 + 2 (1 - sin^{2} (x)) + 5 sin (x)

化简

- 5 + 2 (1 - sin^{2} (x)) + 5 sin (x) : 5 sin (x) - 2 sin^{2} (x) - 3

- 5 + 2 (1 - sin^{2} (x)) + 5 sin (x)

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

数字相乘:

2 \cdot 1 = 2

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

= - 5 + 2 - 2 sin^{2} (x) + 5 sin (x)

数字相加/相减:

- 5 + 2 = - 3

= 5 sin (x) - 2 sin^{2} (x) - 3

= 5 sin (x) - 2 sin^{2} (x) - 3

- 3 - 2 sin^{2} (x) + 5 sin (x) = 0

用替代法求解

- 3 - 2 sin^{2} (x) + 5 sin (x) = 0

令

: sin (x) = u

- 3 - 2 u^{2} + 5 u = 0

- 3 - 2 u^{2} + 5 u = 0 : u = 1, u = \frac{3}{2}

- 3 - 2 u^{2} + 5 u = 0

改写成标准形式

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

- 2 u^{2} + 5 u - 3 = 0

使用求根公式求解

- 2 u^{2} + 5 u - 3 = 0

二次方程求根公式:

若

a = - 2, b = 5, c = - 3

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 2 ) ( - 3 )}{2 ( - 2 )}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 2 ) ( - 3 )}{2 ( - 2 )}

5^{2} - 4 (- 2) (- 3) = 1

5^{2} - 4 (- 2) (- 3)

使用法则

- (- a) = a

= 5^{2} - 4 \cdot 2 \cdot 3

数字相乘:

4 \cdot 2 \cdot 3 = 24

= 5^{2} - 24

5^{2} = 25

= 25 - 24

数字相减:

25 - 24 = 1

= 1

使用法则

1 = 1

= 1

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

将解分隔开

u_{1} = \frac{- 5 + 1}{2 ( - 2 )}, u_{2} = \frac{- 5 - 1}{2 ( - 2 )}

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

\frac{- 5 + 1}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 5 + 1}{- 2 \cdot 2}

数字相加/相减:

- 5 + 1 = - 4

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 4}{- 4}

使用分式法则:

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

= \frac{4}{4}

使用法则

\frac{a}{a} = 1

= 1

u = \frac{- 5 - 1}{2 ( - 2 )} : \frac{3}{2}

\frac{- 5 - 1}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 5 - 1}{- 2 \cdot 2}

数字相减:

- 5 - 1 = - 6

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 6}{- 4}

使用分式法则:

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

= \frac{6}{4}

约分:

2

= \frac{3}{2}

二次方程组的解是:

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

u = sin (x)

代回

sin (x) = 1, sin (x) = \frac{3}{2}

sin (x) = 1, sin (x) = \frac{3}{2}

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

sin (x) = \frac{3}{2} :

无解

sin (x) = \frac{3}{2}

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = \frac{π}{2} + 2 πn

作图

流行的例子

(h(sin^2(x)))/((cos(x)-1))=0

\frac{h ( sin ^{2} ( x ) )}{( cos ( x ) - 1 )} = 0

cos (a) = \frac{- 3}{5}

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

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

sin^{22}(x)= 1/2

sin^{22} (x) = \frac{1}{2}

cos^5(x)=cos(x)

cos^{5} (x) = cos (x)