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受欢迎的三角函数 >

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

解答

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

解答

x = 2 πn, x = π + 2 πn, x = 1.10714 \dots + πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

2 sin (x) cos (x)

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

分解

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

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

因式分解出通项

sin (x)

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

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

分别求解每个部分

sin (x) = 0 or sin (x) - 2 cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0

的通解

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

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sin (x) - 2 cos (x) = 0 : x = arctan (2) + πn

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

使用三角恒等式改写

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

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

tan (x) - 2 = 0

tan (x) - 2 = 0

将

2

到右边

tan (x) - 2 = 0

两边加上

2

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

化简

tan (x) = 2

tan (x) = 2

使用反三角函数性质

tan (x) = 2

tan (x) = 2

的通解

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

x = arctan (2) + πn

x = arctan (2) + πn

合并所有解

x = 2 πn, x = π + 2 πn, x = arctan (2) + πn

以小数形式表示解

x = 2 πn, x = π + 2 πn, x = 1.10714 \dots + πn

作图

流行的例子

cos (x) = 0.77

cos(θ)=-3/(sqrt(5))

cos (θ) = - \frac{3}{5}

2sec(x)+tan^2(x)-9=1

2 sec (x) + tan^{2} (x) - 9 = 1

sin (x) = 1.34

tan(x)sin^2(x)=sin(2x)

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