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受欢迎的三角函数 >

5sin(2x)+3cos(2x)=0,0<= x<= 180

解答

5 sin (2 x) + 3 cos (2 x) = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

解答

x = \frac{- 0.54041 \dots + 18 0 ^{\circ}}{2}, x = \frac{- 0.54041 \dots + 36 0 ^{\circ}}{2}

+1

弧度

x = \frac{- 0.54041 \dots + π}{2}, x = \frac{- 0.54041 \dots + 2 π}{2}

求解步骤

5 sin (2 x) + 3 cos (2 x) = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

使用三角恒等式改写

5 sin (2 x) + 3 cos (2 x) = 0

在两边除以

cos (2 x), cos (2 x) \neq = 0

\frac{5 sin ( 2 x ) + 3 cos ( 2 x )}{cos ( 2 x )} = \frac{0}{cos ( 2 x )}

化简

\frac{5 sin ( 2 x )}{cos ( 2 x )} + 3 = 0

使用基本三角恒等式:

\frac{sin ( x )}{cos ( x )} = tan (x)

5 tan (2 x) + 3 = 0

5 tan (2 x) + 3 = 0

将

3

到右边

5 tan (2 x) + 3 = 0

两边减去

3

5 tan (2 x) + 3 - 3 = 0 - 3

化简

5 tan (2 x) = - 3

5 tan (2 x) = - 3

两边除以

5

5 tan (2 x) = - 3

两边除以

5

\frac{5 tan ( 2 x )}{5} = \frac{- 3}{5}

化简

tan (2 x) = - \frac{3}{5}

tan (2 x) = - \frac{3}{5}

使用反三角函数性质

tan (2 x) = - \frac{3}{5}

tan (2 x) = - \frac{3}{5}

的通解

tan (x) = - a \Rightarrow x = arctan (- a) + 18 0^{\circ} n

2 x = arctan (- \frac{3}{5}) + 18 0^{\circ} n

2 x = arctan (- \frac{3}{5}) + 18 0^{\circ} n

解

2 x = arctan (- \frac{3}{5}) + 18 0^{\circ} n : x = - \frac{arctan ( \frac{3}{5} )}{2} + \frac{18 0 ^{\circ} n}{2}

2 x = arctan (- \frac{3}{5}) + 18 0^{\circ} n

化简

arctan (- \frac{3}{5}) + 18 0^{\circ} n : - arctan (\frac{3}{5}) + 18 0^{\circ} n

arctan (- \frac{3}{5}) + 18 0^{\circ} n

利用以下特性:

arctan (- x) = - arctan (x)

arctan (- \frac{3}{5}) = - arctan (\frac{3}{5})

= - arctan (\frac{3}{5}) + 18 0^{\circ} n

2 x = - arctan (\frac{3}{5}) + 18 0^{\circ} n

两边除以

2

2 x = - arctan (\frac{3}{5}) + 18 0^{\circ} n

两边除以

2

\frac{2 x}{2} = - \frac{arctan ( \frac{3}{5} )}{2} + \frac{18 0 ^{\circ} n}{2}

化简

x = - \frac{arctan ( \frac{3}{5} )}{2} + \frac{18 0 ^{\circ} n}{2}

x = - \frac{arctan ( \frac{3}{5} )}{2} + \frac{18 0 ^{\circ} n}{2}

x = - \frac{arctan ( \frac{3}{5} )}{2} + \frac{18 0 ^{\circ} n}{2}

在

0 \leq x \leq 18 0^{\circ}

范围内的解

x = \frac{- arctan ( \frac{3}{5} ) + 18 0 ^{\circ}}{2}, x = \frac{- arctan ( \frac{3}{5} ) + 36 0 ^{\circ}}{2}

以小数形式表示解

x = \frac{- 0.54041 \dots + 18 0 ^{\circ}}{2}, x = \frac{- 0.54041 \dots + 36 0 ^{\circ}}{2}

作图

流行的例子

cos(θ)=0.6798,0<= θ<= 360

cos (θ) = 0.6798, 0^{\circ} \leq θ \leq 36 0^{\circ}

csc(3θ)=sec(15)

csc (3 θ) = sec (1 5^{\circ})

2cot^2(x)+tan(x)=cot(x)

2 cot^{2} (x) + tan (x) = cot (x)

sin(2x)=sin(pi/2-x)

sin (2 x) = sin (\frac{π}{2} - x)

tan (X) = 1