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受欢迎的三角函数 >

9-12sin(x)=4cos^2(x)

解答

9 - 12 sin (x) = 4 cos^{2} (x)

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

9 - 12 sin (x) = 4 cos^{2} (x)

两边减去

4 cos^{2} (x)

9 - 12 sin (x) - 4 cos^{2} (x) = 0

使用三角恒等式改写

9 - 12 sin (x) - 4 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)

= 9 - 12 sin (x) - 4 (1 - sin^{2} (x))

化简

9 - 12 sin (x) - 4 (1 - sin^{2} (x)) : 4 sin^{2} (x) - 12 sin (x) + 5

9 - 12 sin (x) - 4 (1 - sin^{2} (x))

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

使用加减运算法则

- (- a) = a

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

数字相乘:

4 \cdot 1 = 4

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

= 9 - 12 sin (x) - 4 + 4 sin^{2} (x)

化简

9 - 12 sin (x) - 4 + 4 sin^{2} (x) : 4 sin^{2} (x) - 12 sin (x) + 5

9 - 12 sin (x) - 4 + 4 sin^{2} (x)

对同类项分组

= - 12 sin (x) + 4 sin^{2} (x) + 9 - 4

数字相加/相减:

9 - 4 = 5

= 4 sin^{2} (x) - 12 sin (x) + 5

= 4 sin^{2} (x) - 12 sin (x) + 5

= 4 sin^{2} (x) - 12 sin (x) + 5

5 - 12 sin (x) + 4 sin^{2} (x) = 0

用替代法求解

5 - 12 sin (x) + 4 sin^{2} (x) = 0

令

: sin (x) = u

5 - 12 u + 4 u^{2} = 0

5 - 12 u + 4 u^{2} = 0 : u = \frac{5}{2}, u = \frac{1}{2}

5 - 12 u + 4 u^{2} = 0

改写成标准形式

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

4 u^{2} - 12 u + 5 = 0

使用求根公式求解

4 u^{2} - 12 u + 5 = 0

二次方程求根公式:

若

a = 4, b = - 12, c = 5

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 \cdot 4 \cdot 5}{2 \cdot 4}

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 \cdot 4 \cdot 5}{2 \cdot 4}

(- 12)^{2} - 4 \cdot 4 \cdot 5 = 8

(- 12)^{2} - 4 \cdot 4 \cdot 5

使用指数法则:

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

若

n

是偶数

(- 12)^{2} = 1 2^{2}

= 1 2^{2} - 4 \cdot 4 \cdot 5

数字相乘:

4 \cdot 4 \cdot 5 = 80

= 1 2^{2} - 80

1 2^{2} = 144

= 144 - 80

数字相减:

144 - 80 = 64

= 64

因式分解数字:

64 = 8^{2}

= 8^{2}

使用根式运算法则:

n a^{n} = a

8^{2} = 8

= 8

u_{1, 2} = \frac{- ( - 12 ) \pm 8}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- ( - 12 ) + 8}{2 \cdot 4}, u_{2} = \frac{- ( - 12 ) - 8}{2 \cdot 4}

u = \frac{- ( - 12 ) + 8}{2 \cdot 4} : \frac{5}{2}

\frac{- ( - 12 ) + 8}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{12 + 8}{2 \cdot 4}

数字相加:

12 + 8 = 20

= \frac{20}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{20}{8}

约分:

4

= \frac{5}{2}

u = \frac{- ( - 12 ) - 8}{2 \cdot 4} : \frac{1}{2}

\frac{- ( - 12 ) - 8}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{12 - 8}{2 \cdot 4}

数字相减:

12 - 8 = 4

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

数字相乘:

2 \cdot 4 = 8

= \frac{4}{8}

约分:

4

= \frac{1}{2}

二次方程组的解是:

u = \frac{5}{2}, u = \frac{1}{2}

u = sin (x)

代回

sin (x) = \frac{5}{2}, sin (x) = \frac{1}{2}

sin (x) = \frac{5}{2}, sin (x) = \frac{1}{2}

sin (x) = \frac{5}{2} :

无解

sin (x) = \frac{5}{2}

- 1 \leq sin (x) \leq 1

无解

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

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

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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

作图

流行的例子

(1-sin(x))/(cos(x))=(cos(x))/(1-sin(x))

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

5 sec (x) + 7 = - 3

cos(2θ)+2cos(θ)=-1

cos (2 θ) + 2 cos (θ) = - 1

((tan(x))/1)^{-1}=(20)/(9.7)

(\frac{tan ( x )}{1})^{- 1} = \frac{20}{9.7}

1-tan(x)=sec(x)

1 - tan (x) = sec (x)