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受欢迎的三角函数 >

csc(t)=sec(t)

解答

csc (t) = sec (t)

解答

t = \frac{π}{4} + πn

+1

度数

t = 4 5^{\circ} + 18 0^{\circ} n

求解步骤

csc (t) = sec (t)

两边减去

sec (t)

csc (t) - sec (t) = 0

用 sin, cos 表示

csc (t) - sec (t)

使用基本三角恒等式:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( t )} - sec (t)

使用基本三角恒等式:

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

= \frac{1}{sin ( t )} - \frac{1}{cos ( t )}

化简

\frac{1}{sin ( t )} - \frac{1}{cos ( t )} : \frac{cos ( t ) - sin ( t )}{sin ( t ) cos ( t )}

\frac{1}{sin ( t )} - \frac{1}{cos ( t )}

sin (t), cos (t)

的最小公倍数:

sin (t) cos (t)

sin (t), cos (t)

最小公倍数 (LCM)

计算出由出现在

sin (t)

或

cos (t)

中的因子组成的表达式

= sin (t) cos (t)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

sin (t) cos (t)

对于

\frac{1}{sin ( t )} :

将分母和分子乘以

cos (t)

\frac{1}{sin ( t )} = \frac{1 \cdot cos ( t )}{sin ( t ) cos ( t )} = \frac{cos ( t )}{sin ( t ) cos ( t )}

对于

\frac{1}{cos ( t )} :

将分母和分子乘以

sin (t)

\frac{1}{cos ( t )} = \frac{1 \cdot sin ( t )}{cos ( t ) sin ( t )} = \frac{sin ( t )}{sin ( t ) cos ( t )}

= \frac{cos ( t )}{sin ( t ) cos ( t )} - \frac{sin ( t )}{sin ( t ) cos ( t )}

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{cos ( t ) - sin ( t )}{sin ( t ) cos ( t )}

= \frac{cos ( t ) - sin ( t )}{sin ( t ) cos ( t )}

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

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

cos (t) - sin (t) = 0

使用三角恒等式改写

cos (t) - sin (t) = 0

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

1 - tan (t) = 0

1 - tan (t) = 0

将

1

到右边

1 - tan (t) = 0

两边减去

1

1 - tan (t) - 1 = 0 - 1

化简

- tan (t) = - 1

- tan (t) = - 1

两边除以

- 1

- tan (t) = - 1

两边除以

- 1

\frac{- tan ( t )}{- 1} = \frac{- 1}{- 1}

化简

tan (t) = 1

tan (t) = 1

tan (t) = 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}

t = \frac{π}{4} + πn

t = \frac{π}{4} + πn

作图

流行的例子

-2sin^2(x)=sin(x)-1

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

cos(4θ)-cos(6θ)=0

cos (4 θ) - cos (6 θ) = 0

1 - sin (θ) = 0

9 sin^{2} (θ) - 1 = 0

sinh (x) = 3