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受欢迎的三角函数 >

5cos(x)+5sin(x)tan(x)=10

解答

5 cos (x) + 5 sin (x) tan (x) = 10

解答

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

5 cos (x) + 5 sin (x) tan (x) = 10

两边减去

10

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

使用三角恒等式改写

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

使用基本三角恒等式:

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

= - 10 + 5 cos (x) + 5 sin (x) \frac{sin ( x )}{cos ( x )}

5 sin (x) \frac{sin ( x )}{cos ( x )} = \frac{5 sin ^{2} ( x )}{cos ( x )}

5 sin (x) \frac{sin ( x )}{cos ( x )}

分式相乘:

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

= \frac{sin ( x ) \cdot 5 sin ( x )}{cos ( x )}

sin (x) \cdot 5 sin (x) = 5 sin^{2} (x)

sin (x) \cdot 5 sin (x)

使用指数法则:

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

sin (x) sin (x) = sin^{1 + 1} (x)

= 5 sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= 5 sin^{2} (x)

= \frac{5 sin ^{2} ( x )}{cos ( x )}

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

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= - 10 + \frac{5 ( 1 - cos ^{2} ( x ) )}{cos ( x )} + 5 cos (x)

合并分式

\frac{5 ( - cos ^{2} ( x ) + 1 )}{cos ( x )} + 5 cos (x) : \frac{5}{cos ( x )}

\frac{5 ( - cos ^{2} ( x ) + 1 )}{cos ( x )} + 5 cos (x)

将项转换为分式:

5 cos (x) = \frac{5 cos ( x ) cos ( x )}{cos ( x )}

= \frac{5 ( 1 - cos ^{2} ( x ) )}{cos ( x )} + \frac{5 cos ( x ) cos ( x )}{cos ( x )}

因为分母相等，所以合并分式:

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

= \frac{5 ( 1 - cos ^{2} ( x ) ) + 5 cos ( x ) cos ( x )}{cos ( x )}

5 (1 - cos^{2} (x)) + 5 cos (x) cos (x) = 5 (1 - cos^{2} (x)) + 5 cos^{2} (x)

5 (1 - cos^{2} (x)) + 5 cos (x) cos (x)

5 cos (x) cos (x) = 5 cos^{2} (x)

5 cos (x) cos (x)

使用指数法则:

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

cos (x) cos (x) = cos^{1 + 1} (x)

= 5 cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= 5 cos^{2} (x)

= 5 (- cos^{2} (x) + 1) + 5 cos^{2} (x)

= \frac{5 ( - cos ^{2} ( x ) + 1 ) + 5 cos ^{2} ( x )}{cos ( x )}

乘开

5 (1 - cos^{2} (x)) + 5 cos^{2} (x) : 5

5 (1 - cos^{2} (x)) + 5 cos^{2} (x)

乘开

5 (1 - cos^{2} (x)) : 5 - 5 cos^{2} (x)

5 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 5, b = 1, c = cos^{2} (x)

= 5 \cdot 1 - 5 cos^{2} (x)

数字相乘:

5 \cdot 1 = 5

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

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

同类项相加:

- 5 cos^{2} (x) + 5 cos^{2} (x) = 0

= 5

= \frac{5}{cos ( x )}

= \frac{5}{cos ( x )} - 10

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

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

在两边乘以

cos (x)

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

在两边乘以

cos (x)

- 10 cos (x) + \frac{5}{cos ( x )} cos (x) = 0 \cdot cos (x)

化简

- 10 cos (x) + \frac{5}{cos ( x )} cos (x) = 0 \cdot cos (x)

化简

\frac{5}{cos ( x )} cos (x) : 5

\frac{5}{cos ( x )} cos (x)

分式相乘:

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

= \frac{5 cos ( x )}{cos ( x )}

约分:

cos (x)

= 5

化简

0 \cdot cos (x) : 0

0 \cdot cos (x)

使用法则

0 \cdot a = 0

= 0

- 10 cos (x) + 5 = 0

- 10 cos (x) + 5 = 0

- 10 cos (x) + 5 = 0

将

5

到右边

- 10 cos (x) + 5 = 0

两边减去

5

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

化简

- 10 cos (x) = - 5

- 10 cos (x) = - 5

两边除以

- 10

- 10 cos (x) = - 5

两边除以

- 10

\frac{- 10 cos ( x )}{- 10} = \frac{- 5}{- 10}

化简

\frac{- 10 cos ( x )}{- 10} = \frac{- 5}{- 10}

化简

\frac{- 10 cos ( x )}{- 10} : cos (x)

\frac{- 10 cos ( x )}{- 10}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{10 cos ( x )}{10}

数字相除:

\frac{10}{10} = 1

= cos (x)

化简

\frac{- 5}{- 10} : \frac{1}{2}

\frac{- 5}{- 10}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{5}{10}

约分:

5

= \frac{1}{2}

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

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

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

验证解

找到无定义的点（奇点）:

cos (x) = 0

取

- 10 + \frac{5}{cos ( x )}

的分母，令其等于零

cos (x) = 0

以下点无定义

cos (x) = 0

将不在定义域的点与解相综合:

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

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

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

作图

流行的例子

sin (x) = 0.33

2 cos (2 x) = - 1

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

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

5sin(2θ)-7sin(θ)=0

5 sin (2 θ) - 7 sin (θ) = 0

cos(θ)=(sqrt(3))/3

cos (θ) = \frac{3}{3}