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受欢迎的三角函数 >

sin^2(x)-15sin(x)cos(x)+50cos^2(x)=0

解答

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x) = 0

解答

x = 1.37340 \dots + πn, x = 1.47112 \dots + πn

+1

度数

x = 78.69006 \dots^{\circ} + 18 0^{\circ} n, x = 84.28940 \dots^{\circ} + 18 0^{\circ} n

求解步骤

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x) = 0

分解

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x) : (sin (x) - 5 cos (x)) (sin (x) - 10 cos (x))

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x)

将表达式拆分成组

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x)

定义

50

的因数:

1, 2, 5, 10, 25, 50

50

约数 (因数)

找到

50

的质因数:

2, 5, 5

50

50

除以

250 = 25 \cdot 2

= 2 \cdot 25

25

除以

525 = 5 \cdot 5

= 2 \cdot 5 \cdot 5

2, 5

都是质数，因此无法进一步因数分解

= 2 \cdot 5 \cdot 5

乘以

50

的质因数:

10, 25

2 \cdot 5 = 10

5 \cdot 5 = 25

10, 25

10, 25

添加质因数:

2, 5

将 1 和数字

50

自身相加

1, 50

50

的因数

1, 2, 5, 10, 25, 50

50

的负因数:

- 1, - 2, - 5, - 10, - 25, - 50

将因数乘以

- 1

得到负因数

- 1, - 2, - 5, - 10, - 25, - 50

对于每两个因数

u * v = 50

，检验是否

u + v = - 15

检验

u = 1, v = 50 : u * v = 50, u + v = 51 \Rightarrow

假检验

u = 2, v = 25 : u * v = 50, u + v = 27 \Rightarrow

假

u = - 5, v = - 10

分组为

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

(sin^{2} (x) - 5 sin (x) cos (x)) + (- 10 sin (x) cos (x) + 50 cos^{2} (x))

= (sin^{2} (x) - 5 sin (x) cos (x)) + (- 10 sin (x) cos (x) + 50 cos^{2} (x))

从

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

分解出因式

sin (x)

:

sin (x) (sin (x) - 5 cos (x))

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

使用指数法则:

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

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

= sin (x) sin (x) - 5 sin (x) cos (x)

因式分解出通项

sin (x)

= sin (x) (sin (x) - 5 cos (x))

从

- 10 sin (x) cos (x) + 50 cos^{2} (x)

分解出因式

- 10 cos (x)

:

- 10 cos (x) (sin (x) - 5 cos (x))

- 10 sin (x) cos (x) + 50 cos^{2} (x)

使用指数法则:

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

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

= - 10 sin (x) cos (x) + 50 cos (x) cos (x)

将

50

改写为

5 \cdot 10

= - 10 sin (x) cos (x) + 5 \cdot 10 cos (x) cos (x)

因式分解出通项

- 10 cos (x)

= - 10 cos (x) (sin (x) - 5 cos (x))

= sin (x) (sin (x) - 5 cos (x)) - 10 cos (x) (sin (x) - 5 cos (x))

因式分解出通项

sin (x) - 5 cos (x)

= (sin (x) - 5 cos (x)) (sin (x) - 10 cos (x))

(sin (x) - 5 cos (x)) (sin (x) - 10 cos (x)) = 0

分别求解每个部分

sin (x) - 5 cos (x) = 0 or sin (x) - 10 cos (x) = 0

sin (x) - 5 cos (x) = 0 : x = arctan (5) + πn

sin (x) - 5 cos (x) = 0

使用三角恒等式改写

sin (x) - 5 cos (x) = 0

在两边除以

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

\frac{sin ( x ) - 5 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{sin ( x )}{cos ( x )} - 5 = 0

使用基本三角恒等式:

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

tan (x) - 5 = 0

tan (x) - 5 = 0

将

5

到右边

tan (x) - 5 = 0

两边加上

5

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

化简

tan (x) = 5

tan (x) = 5

使用反三角函数性质

tan (x) = 5

tan (x) = 5

的通解

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

x = arctan (5) + πn

x = arctan (5) + πn

sin (x) - 10 cos (x) = 0 : x = arctan (10) + πn

sin (x) - 10 cos (x) = 0

使用三角恒等式改写

sin (x) - 10 cos (x) = 0

在两边除以

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

\frac{sin ( x ) - 10 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{sin ( x )}{cos ( x )} - 10 = 0

使用基本三角恒等式:

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

tan (x) - 10 = 0

tan (x) - 10 = 0

将

10

到右边

tan (x) - 10 = 0

两边加上

10

tan (x) - 10 + 10 = 0 + 10

化简

tan (x) = 10

tan (x) = 10

使用反三角函数性质

tan (x) = 10

tan (x) = 10

的通解

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

x = arctan (10) + πn

x = arctan (10) + πn

合并所有解

x = arctan (5) + πn, x = arctan (10) + πn

以小数形式表示解

x = 1.37340 \dots + πn, x = 1.47112 \dots + πn

作图

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