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受欢迎的三角函数 >

sec^2(θ)tan(θ)=2tan(θ),0<= θ<= 2pi

解答

sec^{2} (θ) tan (θ) = 2 tan (θ), 0 \leq θ \leq 2 π

解答

θ = 0, θ = π, θ = 2 π, θ = \frac{3 π}{4}, θ = \frac{5 π}{4}, θ = \frac{π}{4}, θ = \frac{7 π}{4}

+1

度数

θ = 0^{\circ}, θ = 18 0^{\circ}, θ = 36 0^{\circ}, θ = 13 5^{\circ}, θ = 22 5^{\circ}, θ = 4 5^{\circ}, θ = 31 5^{\circ}

求解步骤

sec^{2} (θ) tan (θ) = 2 tan (θ), 0 \leq θ \leq 2 π

两边减去

2 tan (θ)

sec^{2} (θ) tan (θ) - 2 tan (θ) = 0

分解

sec^{2} (θ) tan (θ) - 2 tan (θ) : tan (θ) (sec (θ) + 2) (sec (θ) - 2)

sec^{2} (θ) tan (θ) - 2 tan (θ)

因式分解出通项

tan (θ)

= tan (θ) (sec^{2} (θ) - 2)

分解

sec^{2} (θ) - 2 : (sec (θ) + 2) (sec (θ) - 2)

sec^{2} (θ) - 2

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

= sec^{2} (θ) - (2)^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

sec^{2} (θ) - (2)^{2} = (sec (θ) + 2) (sec (θ) - 2)

= (sec (θ) + 2) (sec (θ) - 2)

= tan (θ) (sec (θ) + 2) (sec (θ) - 2)

tan (θ) (sec (θ) + 2) (sec (θ) - 2) = 0

分别求解每个部分

tan (θ) = 0 or sec (θ) + 2 = 0 or sec (θ) - 2 = 0

tan (θ) = 0, 0 \leq θ \leq 2 π : θ = 0, θ = π, θ = 2 π

tan (θ) = 0, 0 \leq θ \leq 2 π

tan (θ) = 0

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}

θ = 0 + πn

θ = 0 + πn

解

θ = 0 + πn : θ = πn

θ = 0 + πn

0 + πn = πn

θ = πn

θ = πn

在

0 \leq θ \leq 2 π

范围内的解

θ = 0, θ = π, θ = 2 π

sec (θ) + 2 = 0, 0 \leq θ \leq 2 π : θ = \frac{3 π}{4}, θ = \frac{5 π}{4}

sec (θ) + 2 = 0, 0 \leq θ \leq 2 π

将

2

到右边

sec (θ) + 2 = 0

两边减去

2

sec (θ) + 2 - 2 = 0 - 2

化简

sec (θ) = - 2

sec (θ) = - 2

sec (θ) = - 2

的通解

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

在

0 \leq θ \leq 2 π

范围内的解

θ = \frac{3 π}{4}, θ = \frac{5 π}{4}

sec (θ) - 2 = 0, 0 \leq θ \leq 2 π : θ = \frac{π}{4}, θ = \frac{7 π}{4}

sec (θ) - 2 = 0, 0 \leq θ \leq 2 π

将

2

到右边

sec (θ) - 2 = 0

两边加上

2

sec (θ) - 2 + 2 = 0 + 2

化简

sec (θ) = 2

sec (θ) = 2

sec (θ) = 2

的通解

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

在

0 \leq θ \leq 2 π

范围内的解

θ = \frac{π}{4}, θ = \frac{7 π}{4}

合并所有解

θ = 0, θ = π, θ = 2 π, θ = \frac{3 π}{4}, θ = \frac{5 π}{4}, θ = \frac{π}{4}, θ = \frac{7 π}{4}

作图

流行的例子

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

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

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

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

250sin(75)=393.19sin(45-θ)

250 sin (7 5^{\circ}) = 393.19 sin (4 5^{\circ} - θ)

tan(θ)= 6/(6.71)

tan (θ) = \frac{6}{6.71}

sin (2 θ) = 0.5