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受欢迎的三角函数 >

6cos(2x)=6cos^2(x)-5

解答

6 cos (2 x) = 6 cos^{2} (x) - 5

解答

x = 1.15026 \dots + 2 πn, x = π - 1.15026 \dots + 2 πn, x = - 1.15026 \dots + 2 πn, x = π + 1.15026 \dots + 2 πn

+1

度数

x = 65.90515 \dots^{\circ} + 36 0^{\circ} n, x = 114.09484 \dots^{\circ} + 36 0^{\circ} n, x = - 65.90515 \dots^{\circ} + 36 0^{\circ} n, x = 245.90515 \dots^{\circ} + 36 0^{\circ} n

求解步骤

6 cos (2 x) = 6 cos^{2} (x) - 5

两边减去

6 cos^{2} (x) - 5

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

使用三角恒等式改写

5 + 6 cos (2 x) - 6 cos^{2} (x)

使用倍角公式:

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

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

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

化简

5 + 6 cos (2 x) - 6 (cos (2 x) + sin^{2} (x)) : 5 - 6 sin^{2} (x)

5 + 6 cos (2 x) - 6 (cos (2 x) + sin^{2} (x))

乘开

- 6 (cos (2 x) + sin^{2} (x)) : - 6 cos (2 x) - 6 sin^{2} (x)

- 6 (cos (2 x) + sin^{2} (x))

使用分配律:

a (b + c) = ab + a c

a = - 6, b = cos (2 x), c = sin^{2} (x)

= - 6 cos (2 x) + (- 6) sin^{2} (x)

使用加减运算法则

+ (- a) = - a

= - 6 cos (2 x) - 6 sin^{2} (x)

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

同类项相加:

6 cos (2 x) - 6 cos (2 x) = 0

= 5 - 6 sin^{2} (x)

= 5 - 6 sin^{2} (x)

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

用替代法求解

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

令

: sin (x) = u

5 - 6 u^{2} = 0

5 - 6 u^{2} = 0 : u = \frac{5}{6}, u = - \frac{5}{6}

5 - 6 u^{2} = 0

将

5

到右边

5 - 6 u^{2} = 0

两边减去

5

5 - 6 u^{2} - 5 = 0 - 5

化简

- 6 u^{2} = - 5

- 6 u^{2} = - 5

两边除以

- 6

- 6 u^{2} = - 5

两边除以

- 6

\frac{- 6 u ^{2}}{- 6} = \frac{- 5}{- 6}

化简

u^{2} = \frac{5}{6}

u^{2} = \frac{5}{6}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{5}{6}, u = - \frac{5}{6}

u = sin (x)

代回

sin (x) = \frac{5}{6}, sin (x) = - \frac{5}{6}

sin (x) = \frac{5}{6}, sin (x) = - \frac{5}{6}

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

sin (x) = \frac{5}{6}

使用反三角函数性质

sin (x) = \frac{5}{6}

sin (x) = \frac{5}{6}

的通解

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

x = arcsin (\frac{5}{6}) + 2 πn, x = π - arcsin (\frac{5}{6}) + 2 πn

x = arcsin (\frac{5}{6}) + 2 πn, x = π - arcsin (\frac{5}{6}) + 2 πn

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

sin (x) = - \frac{5}{6}

使用反三角函数性质

sin (x) = - \frac{5}{6}

sin (x) = - \frac{5}{6}

的通解

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

x = arcsin (- \frac{5}{6}) + 2 πn, x = π + arcsin (\frac{5}{6}) + 2 πn

x = arcsin (- \frac{5}{6}) + 2 πn, x = π + arcsin (\frac{5}{6}) + 2 πn

合并所有解

x = arcsin (\frac{5}{6}) + 2 πn, x = π - arcsin (\frac{5}{6}) + 2 πn, x = arcsin (- \frac{5}{6}) + 2 πn, x = π + arcsin (\frac{5}{6}) + 2 πn

以小数形式表示解

x = 1.15026 \dots + 2 πn, x = π - 1.15026 \dots + 2 πn, x = - 1.15026 \dots + 2 πn, x = π + 1.15026 \dots + 2 πn

作图

流行的例子

(-4cos(x)-6sin(x))^2-20sin^2(x)=16

(- 4 cos (x) - 6 sin (x))^{2} - 20 sin^{2} (x) = 16

sin (u) = \frac{5}{13}

0.8 = cos (x)

cos (x) = \frac{12}{20}

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

2 sin^{2} (x) + sin (x) = 1, - π \leq x \leq π