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受欢迎的三角函数 >

64cos^2(θ)=49,(0<= ,θ<= 360)

解答

64 cos^{2} (θ) = 49, (0^{\circ} \leq, θ \leq 36 0^{\circ})

解答

θ \in R 无解

求解步骤

64 cos^{2} (θ) = 49, (0^{\circ} \leq, θ \leq 36 0^{\circ})

用替代法求解

64 cos^{2} (θ) = 49

令

: cos (θ) = u

64 u^{2} = 49

64 u^{2} = 49 : u = \frac{7}{8}, u = - \frac{7}{8}

64 u^{2} = 49

两边除以

64

64 u^{2} = 49

两边除以

64

\frac{64 u ^{2}}{64} = \frac{49}{64}

化简

u^{2} = \frac{49}{64}

u^{2} = \frac{49}{64}

对于

x^{2} = f (a)

解为

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

u = \frac{49}{64}, u = - \frac{49}{64}

\frac{49}{64} = \frac{7}{8}

\frac{49}{64}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{49}{64}

64 = 8

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

= \frac{49}{8}

49 = 7

49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

= \frac{7}{8}

- \frac{49}{64} = - \frac{7}{8}

- \frac{49}{64}

化简

\frac{49}{64} : \frac{7}{8}

\frac{49}{64}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{49}{64}

64 = 8

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

= \frac{49}{8}

49 = 7

49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

= \frac{7}{8}

= - \frac{7}{8}

u = \frac{7}{8}, u = - \frac{7}{8}

u = cos (θ)

代回

cos (θ) = \frac{7}{8}, cos (θ) = - \frac{7}{8}

cos (θ) = \frac{7}{8}, cos (θ) = - \frac{7}{8}

cos (θ) = \frac{7}{8}, (0 \leq, θ \leq 36 0^{\circ}) :

无解

cos (θ) = \frac{7}{8}, (0 \leq, θ \leq 36 0^{\circ})

使用反三角函数性质

cos (θ) = \frac{7}{8}

cos (θ) = \frac{7}{8}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

θ = arccos (\frac{7}{8}) + 36 0^{\circ} n, θ = 36 0^{\circ} - arccos (\frac{7}{8}) + 36 0^{\circ} n

θ = arccos (\frac{7}{8}) + 36 0^{\circ} n, θ = 36 0^{\circ} - arccos (\frac{7}{8}) + 36 0^{\circ} n

在

(0 \leq, θ \leq 36 0^{\circ})

范围内的解

无解

cos (θ) = - \frac{7}{8}, (0 \leq, θ \leq 36 0^{\circ}) :

无解

cos (θ) = - \frac{7}{8}, (0 \leq, θ \leq 36 0^{\circ})

使用反三角函数性质

cos (θ) = - \frac{7}{8}

cos (θ) = - \frac{7}{8}

的通解

cos (x) = - a \Rightarrow x = arccos (- a) + 36 0^{\circ} n, x = - arccos (- a) + 36 0^{\circ} n

θ = arccos (- \frac{7}{8}) + 36 0^{\circ} n, θ = - arccos (- \frac{7}{8}) + 36 0^{\circ} n

θ = arccos (- \frac{7}{8}) + 36 0^{\circ} n, θ = - arccos (- \frac{7}{8}) + 36 0^{\circ} n

在

(0 \leq, θ \leq 36 0^{\circ})

范围内的解

无解

合并所有解

θ \in R 无解

作图

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