zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

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

解答

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

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

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

两边减去

2

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

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

化简

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

- 2 + cos (x) + sin (x) \frac{sin ( x )}{cos ( x )}

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

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

分式相乘:

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

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

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

sin (x) sin (x)

使用指数法则:

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

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

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

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

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

将项转换为分式:

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

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

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

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

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

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)

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

= - 2 cos (x) + 1

- 2 cos (x) + 1 = 0

将

1

到右边

- 2 cos (x) + 1 = 0

两边减去

1

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

化简

- 2 cos (x) = - 1

- 2 cos (x) = - 1

两边除以

- 2

- 2 cos (x) = - 1

两边除以

- 2

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

化简

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

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

tan(θ)=(sqrt(3))/3

tan (θ) = \frac{3}{3}

cos(x/2)=(sqrt(2))/2

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

sin (10 5^{\circ})

3tan^3(θ)=tan(θ)

3 tan^{3} (θ) = tan (θ)

csc^{2} (x) - 4 = 0