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受欢迎的三角函数 >

2sin^2(x)-3sin(x)cos(x)+cos^2(x)=0

解答

2 sin^{2} (x) - 3 sin (x) cos (x) + cos^{2} (x) = 0

解答

x = 0.46364 \dots + πn, x = \frac{π}{4} + πn

+1

度数

x = 26.56505 \dots^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

求解步骤

2 sin^{2} (x) - 3 sin (x) cos (x) + cos^{2} (x) = 0

分解

2 sin^{2} (x) - 3 sin (x) cos (x) + cos^{2} (x) : (2 sin (x) - cos (x)) (sin (x) - cos (x))

2 sin^{2} (x) - 3 sin (x) cos (x) + cos^{2} (x)

将表达式拆分成组

2 sin^{2} (x) - 3 sin (x) cos (x) + cos^{2} (x)

定义

2

的因数:

1, 2

2

约数 (因数)

找到

2

的质因数:

2

2

2

是质数，因此无法因数分解

= 2

加 1

1

2

的因数

1, 2

2

的负因数:

- 1, - 2

将因数乘以

- 1

得到负因数

- 1, - 2

对于每两个因数

u * v = 2

，检验是否

u + v = - 3

检验

u = 1, v = 2 : u * v = 2, u + v = 3 \Rightarrow

假检验

u = - 1, v = - 2 : u * v = 2, u + v = - 3 \Rightarrow

真

u = - 1, v = - 2

分组为

(a x^{2} + ux y) + (vx y + c y^{2})

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

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

从

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

分解出因式

sin (x)

:

sin (x) (2 sin (x) - cos (x))

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

使用指数法则:

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

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

= 2 sin (x) sin (x) - sin (x) cos (x)

因式分解出通项

sin (x)

= sin (x) (2 sin (x) - cos (x))

从

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

分解出因式

- cos (x)

:

- cos (x) (2 sin (x) - cos (x))

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

使用指数法则:

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

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

= - 2 sin (x) cos (x) + cos (x) cos (x)

因式分解出通项

- cos (x)

= - cos (x) (2 sin (x) - cos (x))

= sin (x) (2 sin (x) - cos (x)) - cos (x) (2 sin (x) - cos (x))

因式分解出通项

2 sin (x) - cos (x)

= (2 sin (x) - cos (x)) (sin (x) - cos (x))

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

分别求解每个部分

2 sin (x) - cos (x) = 0 or sin (x) - cos (x) = 0

2 sin (x) - cos (x) = 0 : x = arctan (\frac{1}{2}) + πn

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

使用三角恒等式改写

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

在两边除以

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

\frac{2 sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{2 sin ( x )}{cos ( x )} - 1 = 0

使用基本三角恒等式:

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

2 tan (x) - 1 = 0

2 tan (x) - 1 = 0

将

1

到右边

2 tan (x) - 1 = 0

两边加上

1

2 tan (x) - 1 + 1 = 0 + 1

化简

2 tan (x) = 1

2 tan (x) = 1

两边除以

2

2 tan (x) = 1

两边除以

2

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

化简

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

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

使用反三角函数性质

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

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

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{1}{2}) + πn

x = arctan (\frac{1}{2}) + πn

sin (x) - cos (x) = 0 : x = \frac{π}{4} + πn

sin (x) - cos (x) = 0

使用三角恒等式改写

sin (x) - cos (x) = 0

在两边除以

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

\frac{sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{sin ( x )}{cos ( x )} - 1 = 0

使用基本三角恒等式:

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

tan (x) - 1 = 0

tan (x) - 1 = 0

将

1

到右边

tan (x) - 1 = 0

两边加上

1

tan (x) - 1 + 1 = 0 + 1

化简

tan (x) = 1

tan (x) = 1

tan (x) = 1

的通解

tan (x)

周期表（周期为

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

合并所有解

x = arctan (\frac{1}{2}) + πn, x = \frac{π}{4} + πn

以小数形式表示解

x = 0.46364 \dots + πn, x = \frac{π}{4} + πn

作图

流行的例子

sin (b) = \frac{3}{5}

(tan(θ)-2)(16sin^2(θ)-1)=0

(tan (θ) - 2) (16 sin^{2} (θ) - 1) = 0

csc (θ) = 1.9

cos(x-pi/4)=(sqrt(3))/2

cos (x - \frac{π}{4}) = \frac{3}{2}

cos(x)tan(x)-cos(x)=0,0<= x<= 2pi

cos (x) tan (x) - cos (x) = 0, 0 \leq x \leq 2 π