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受欢迎的三角函数 >

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

解答

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

解答

x = - 0.95531 \dots + πn, x = 0.95531 \dots + πn

+1

度数

x = - 54.73561 \dots^{\circ} + 18 0^{\circ} n, x = 54.73561 \dots^{\circ} + 18 0^{\circ} n

求解步骤

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

分解

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

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

将

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

改写为

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

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

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

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

使用指数法则:

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

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

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

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

使用平方差公式:

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

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

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

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

分别求解每个部分

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

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

两边减去

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

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

以小数形式表示解

x = - 0.95531 \dots + πn, x = 0.95531 \dots + πn

作图

流行的例子

sin(x)= 1/(sqrt(10))

sin (x) = \frac{1}{10}

5 sec (θ) - 6 = 0

sec(x)=-1,-2pi<= x<= 2pi

sec (x) = - 1, - 2 π \leq x \leq 2 π

1/(cos(2x))+tan(2x)=3cos(2x)

\frac{1}{cos ( 2 x )} + tan (2 x) = 3 cos (2 x)

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

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