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受欢迎的三角函数 >

sin^2(x)-3cos^2(x)=0

解答

sin^{2} (x) - 3 cos^{2} (x) = 0

解答

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

+1

度数

x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

求解步骤

sin^{2} (x) - 3 cos^{2} (x) = 0

分解

sin^{2} (x) - 3 cos^{2} (x) : (sin (x) + 3 cos (x)) (sin (x) - 3 cos (x))

sin^{2} (x) - 3 cos^{2} (x)

将

sin^{2} (x) - 3 cos^{2} (x)

改写为

sin^{2} (x) - (3 cos (x))^{2}

sin^{2} (x) - 3 cos^{2} (x)

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

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

使用指数法则:

a^{m} b^{m} = (ab)^{m}

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

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

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

使用平方差公式:

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

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

= (sin (x) + 3 cos (x)) (sin (x) - 3 cos (x))

(sin (x) + 3 cos (x)) (sin (x) - 3 cos (x)) = 0

分别求解每个部分

sin (x) + 3 cos (x) = 0 or sin (x) - 3 cos (x) = 0

sin (x) + 3 cos (x) = 0 : x = \frac{2 π}{3} + πn

sin (x) + 3 cos (x) = 0

使用三角恒等式改写

sin (x) + 3 cos (x) = 0

在两边除以

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

\frac{sin ( x ) + 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{sin ( x )}{cos ( x )} + 3 = 0

使用基本三角恒等式:

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

tan (x) + 3 = 0

tan (x) + 3 = 0

将

3

到右边

tan (x) + 3 = 0

两边减去

3

tan (x) + 3 - 3 = 0 - 3

化简

tan (x) = - 3

tan (x) = - 3

tan (x) = - 3

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}

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

sin (x) - 3 cos (x) = 0 : x = \frac{π}{3} + πn

sin (x) - 3 cos (x) = 0

使用三角恒等式改写

sin (x) - 3 cos (x) = 0

在两边除以

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

\frac{sin ( x ) - 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{sin ( x )}{cos ( x )} - 3 = 0

使用基本三角恒等式:

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

tan (x) - 3 = 0

tan (x) - 3 = 0

将

3

到右边

tan (x) - 3 = 0

两边加上

3

tan (x) - 3 + 3 = 0 + 3

化简

tan (x) = 3

tan (x) = 3

tan (x) = 3

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}

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

合并所有解

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

作图

流行的例子

cos(2x+pi/3)=sin(3x)

cos (2 x + \frac{π}{3}) = sin (3 x)

solvefor y,x=arcsin(y)

so l v e f or y, x = arcsin (y)

2sin^2(x)+5sin(x)=3

2 sin^{2} (x) + 5 sin (x) = 3

∣ sin (x) ∣ = \frac{1}{2}

sec^2(θ)=2tan(θ)

sec^{2} (θ) = 2 tan (θ)