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受欢迎的三角函数 >

4sin(θ)+3csc(θ)=7

解答

4 sin (θ) + 3 csc (θ) = 7

解答

θ = 0.84806 \dots + 2 πn, θ = π - 0.84806 \dots + 2 πn, θ = \frac{π}{2} + 2 πn

+1

度数

θ = 48.59037 \dots^{\circ} + 36 0^{\circ} n, θ = 131.40962 \dots^{\circ} + 36 0^{\circ} n, θ = 9 0^{\circ} + 36 0^{\circ} n

求解步骤

4 sin (θ) + 3 csc (θ) = 7

两边减去

7

4 sin (θ) + 3 csc (θ) - 7 = 0

使用三角恒等式改写

- 7 + 3 csc (θ) + 4 sin (θ)

使用基本三角恒等式:

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

= - 7 + 3 csc (θ) + 4 \cdot \frac{1}{csc ( θ )}

4 \cdot \frac{1}{csc ( θ )} = \frac{4}{csc ( θ )}

4 \cdot \frac{1}{csc ( θ )}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 4}{csc ( θ )}

数字相乘:

1 \cdot 4 = 4

= \frac{4}{csc ( θ )}

= - 7 + 3 csc (θ) + \frac{4}{csc ( θ )}

- 7 + \frac{4}{csc ( θ )} + 3 csc (θ) = 0

用替代法求解

- 7 + \frac{4}{csc ( θ )} + 3 csc (θ) = 0

令

: csc (θ) = u

- 7 + \frac{4}{u} + 3 u = 0

- 7 + \frac{4}{u} + 3 u = 0 : u = \frac{4}{3}, u = 1

- 7 + \frac{4}{u} + 3 u = 0

在两边乘以

u

- 7 + \frac{4}{u} + 3 u = 0

在两边乘以

u

- 7 u + \frac{4}{u} u + 3 uu = 0 \cdot u

化简

- 7 u + \frac{4}{u} u + 3 uu = 0 \cdot u

化简

\frac{4}{u} u : 4

\frac{4}{u} u

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{4 u}{u}

约分:

u

= 4

化简

3 uu : 3 u^{2}

3 uu

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= 3 u^{1 + 1}

数字相加:

1 + 1 = 2

= 3 u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

- 7 u + 4 + 3 u^{2} = 0

- 7 u + 4 + 3 u^{2} = 0

- 7 u + 4 + 3 u^{2} = 0

解

- 7 u + 4 + 3 u^{2} = 0 : u = \frac{4}{3}, u = 1

- 7 u + 4 + 3 u^{2} = 0

改写成标准形式

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

3 u^{2} - 7 u + 4 = 0

使用求根公式求解

3 u^{2} - 7 u + 4 = 0

二次方程求根公式:

若

a = 3, b = - 7, c = 4

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

(- 7)^{2} - 4 \cdot 3 \cdot 4 = 1

(- 7)^{2} - 4 \cdot 3 \cdot 4

使用指数法则:

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

若

n

是偶数

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 3 \cdot 4

数字相乘:

4 \cdot 3 \cdot 4 = 48

= 7^{2} - 48

7^{2} = 49

= 49 - 48

数字相减:

49 - 48 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- ( - 7 ) \pm 1}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 7 ) + 1}{2 \cdot 3}, u_{2} = \frac{- ( - 7 ) - 1}{2 \cdot 3}

u = \frac{- ( - 7 ) + 1}{2 \cdot 3} : \frac{4}{3}

\frac{- ( - 7 ) + 1}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{7 + 1}{2 \cdot 3}

数字相加:

7 + 1 = 8

= \frac{8}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{8}{6}

约分:

2

= \frac{4}{3}

u = \frac{- ( - 7 ) - 1}{2 \cdot 3} : 1

\frac{- ( - 7 ) - 1}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{7 - 1}{2 \cdot 3}

数字相减:

7 - 1 = 6

= \frac{6}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{6}{6}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

u = \frac{4}{3}, u = 1

u = \frac{4}{3}, u = 1

验证解

找到无定义的点（奇点）:

u = 0

取

- 7 + \frac{4}{u} + 3 u

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = \frac{4}{3}, u = 1

u = csc (θ)

代回

csc (θ) = \frac{4}{3}, csc (θ) = 1

csc (θ) = \frac{4}{3}, csc (θ) = 1

csc (θ) = \frac{4}{3} : θ = arccsc (\frac{4}{3}) + 2 πn, θ = π - arccsc (\frac{4}{3}) + 2 πn

csc (θ) = \frac{4}{3}

使用反三角函数性质

csc (θ) = \frac{4}{3}

csc (θ) = \frac{4}{3}

的通解

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

θ = arccsc (\frac{4}{3}) + 2 πn, θ = π - arccsc (\frac{4}{3}) + 2 πn

θ = arccsc (\frac{4}{3}) + 2 πn, θ = π - arccsc (\frac{4}{3}) + 2 πn

csc (θ) = 1 : θ = \frac{π}{2} + 2 πn

csc (θ) = 1

csc (θ) = 1

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

θ = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

合并所有解

θ = arccsc (\frac{4}{3}) + 2 πn, θ = π - arccsc (\frac{4}{3}) + 2 πn, θ = \frac{π}{2} + 2 πn

以小数形式表示解

θ = 0.84806 \dots + 2 πn, θ = π - 0.84806 \dots + 2 πn, θ = \frac{π}{2} + 2 πn

作图

流行的例子

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

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sin^{2} (θ) + cos (θ) (1 - cos (θ)) = 0

tan (A) = \frac{94}{67}

2tan(2x)-10=34641

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