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受欢迎的三角函数 >

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

解答

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

解答

x = 0.30760 \dots + 2 πn, x = π - 0.30760 \dots + 2 πn

+1

度数

x = 17.62439 \dots^{\circ} + 36 0^{\circ} n, x = 162.37560 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

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

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

用替代法求解

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

令

: sin (x) = u

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

1 - u^{2} - 3 u = 0 : u = - \frac{3 + 13}{2}, u = \frac{13 - 3}{2}

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

改写成标准形式

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

- u^{2} - 3 u + 1 = 0

使用求根公式求解

- u^{2} - 3 u + 1 = 0

二次方程求根公式:

若

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

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

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

(- 3)^{2} - 4 (- 1) \cdot 1 = 13

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

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

= 3^{2} + 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 3^{2} + 4

3^{2} = 9

= 9 + 4

数字相加:

9 + 4 = 13

= 13

u_{1, 2} = \frac{- ( - 3 ) \pm 13}{2 ( - 1 )}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 13}{2 ( - 1 )}, u_{2} = \frac{- ( - 3 ) - 13}{2 ( - 1 )}

u = \frac{- ( - 3 ) + 13}{2 ( - 1 )} : - \frac{3 + 13}{2}

\frac{- ( - 3 ) + 13}{2 ( - 1 )}

去除括号:

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

= \frac{3 + 13}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{3 + 13}{- 2}

使用分式法则:

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

= - \frac{3 + 13}{2}

u = \frac{- ( - 3 ) - 13}{2 ( - 1 )} : \frac{13 - 3}{2}

\frac{- ( - 3 ) - 13}{2 ( - 1 )}

去除括号:

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

= \frac{3 - 13}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{3 - 13}{- 2}

使用分式法则:

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

3 - 13 = - (13 - 3)

= \frac{13 - 3}{2}

二次方程组的解是:

u = - \frac{3 + 13}{2}, u = \frac{13 - 3}{2}

u = sin (x)

代回

sin (x) = - \frac{3 + 13}{2}, sin (x) = \frac{13 - 3}{2}

sin (x) = - \frac{3 + 13}{2}, sin (x) = \frac{13 - 3}{2}

sin (x) = - \frac{3 + 13}{2} :

无解

sin (x) = - \frac{3 + 13}{2}

- 1 \leq sin (x) \leq 1

无解

sin (x) = \frac{13 - 3}{2} : x = arcsin (\frac{13 - 3}{2}) + 2 πn, x = π - arcsin (\frac{13 - 3}{2}) + 2 πn

sin (x) = \frac{13 - 3}{2}

使用反三角函数性质

sin (x) = \frac{13 - 3}{2}

sin (x) = \frac{13 - 3}{2}

的通解

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

x = arcsin (\frac{13 - 3}{2}) + 2 πn, x = π - arcsin (\frac{13 - 3}{2}) + 2 πn

x = arcsin (\frac{13 - 3}{2}) + 2 πn, x = π - arcsin (\frac{13 - 3}{2}) + 2 πn

合并所有解

x = arcsin (\frac{13 - 3}{2}) + 2 πn, x = π - arcsin (\frac{13 - 3}{2}) + 2 πn

以小数形式表示解

x = 0.30760 \dots + 2 πn, x = π - 0.30760 \dots + 2 πn

作图

流行的例子

csc(t)= 1/(-(sqrt(2))/2)

csc (t) = \frac{1}{- \frac{2}{2}}

cos^2(x)=1-sin(x),0<= x<2pi

cos^{2} (x) = 1 - sin (x), 0 \leq x < 2 π

cos(2x)-3sin(x)=2,0<= x<= 360

cos (2 x) - 3 sin (x) = 2, 0^{\circ} \leq x \leq 36 0^{\circ}

2cos^2(x)+3cos(x)=4cos(x)+1

2 cos^{2} (x) + 3 cos (x) = 4 cos (x) + 1

6sin^2(x)+5sin(x)=0

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