三角学示例

{sin}^{3} (t) + {cos}^{3} (t) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t) = sin (t) + cos (t)

$\sin^{3} (t) + \cos^{3} (t) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t) = \sin (t) + \cos (t)$

解题步骤 1

从左边开始。

{sin}^{3} (t) + {cos}^{3} (t) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$\sin^{3} (t) + \cos^{3} (t) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

解题步骤 2

因数。

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解题步骤 2.1

因为两项都是完全立方数，所以使用立方和公式

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})

$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ 进行因式分解，其中

a = sin (t)

$a = \sin (t)$ 和

b = cos (t)

$b = \cos (t)$ 。

(sin (t) + cos (t)) ({sin}^{2} (t) - sin (t) cos (t) + {cos}^{2} (t)) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$(\sin (t) + \cos (t)) (\sin^{2} (t) - \sin (t) \cos (t) + \cos^{2} (t)) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

解题步骤 2.2

化简。

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解题步骤 2.2.1

重新整理项。

(sin (t) + cos (t)) (- sin (t) cos (t) + {sin}^{2} (t) + {cos}^{2} (t)) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$(\sin (t) + \cos (t)) (- \sin (t) \cos (t) + \sin^{2} (t) + \cos^{2} (t)) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

解题步骤 2.2.2

使用勾股恒等式。

(sin (t) + cos (t)) (- sin (t) cos (t) + 1) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$(\sin (t) + \cos (t)) (- \sin (t) \cos (t) + 1) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

(sin (t) + cos (t)) (- sin (t) cos (t) + 1) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$(\sin (t) + \cos (t)) (- \sin (t) \cos (t) + 1) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

(sin (t) + cos (t)) (- sin (t) cos (t) + 1) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$(\sin (t) + \cos (t)) (- \sin (t) \cos (t) + 1) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

解题步骤 3

化简。

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解题步骤 3.1

化简每一项。

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解题步骤 3.1.1

使用 FOIL 方法展开

(sin (t) + cos (t)) (- sin (t) cos (t) + 1)

$(\sin (t) + \cos (t)) (- \sin (t) \cos (t) + 1)$ 。

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解题步骤 3.1.1.1

运用分配律。

sin (t) (- sin (t) cos (t) + 1) + cos (t) (- sin (t) cos (t) + 1) + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$\sin (t) (- \sin (t) \cos (t) + 1) + \cos (t) (- \sin (t) \cos (t) + 1) + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.1.2

运用分配律。

sin (t) (- sin (t) cos (t)) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t) + 1) + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$\sin (t) (- \sin (t) \cos (t)) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t) + 1) + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.1.3

运用分配律。

sin (t) (- sin (t) cos (t)) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$\sin (t) (- \sin (t) \cos (t)) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

sin (t) (- sin (t) cos (t)) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$\sin (t) (- \sin (t) \cos (t)) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.2

化简每一项。

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解题步骤 3.1.2.1

乘以

sin (t) (- sin (t) cos (t))

$\sin (t) (- \sin (t) \cos (t))$ 。

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解题步骤 3.1.2.1.1

对

sin (t)

$\sin (t)$ 进行

1

$1$ 次方运算。

- ({sin (t)}^{1} sin (t)) cos (t) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- ({\sin (t)}^{1} \sin (t)) \cos (t) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.2.1.2

对

sin (t)

$\sin (t)$ 进行

1

$1$ 次方运算。

- ({sin (t)}^{1} {sin (t)}^{1}) cos (t) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- ({\sin (t)}^{1} {\sin (t)}^{1}) \cos (t) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.2.1.3

使用幂法则

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ 合并指数。

- {sin (t)}^{1 + 1} cos (t) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{1 + 1} \cos (t) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.2.1.4

将

1

$1$ 和

1

$1$ 相加。

- {sin (t)}^{2} cos (t) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

- {sin (t)}^{2} cos (t) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.2.2

将

sin (t)

$\sin (t)$ 乘以

1

$1$ 。

- {sin (t)}^{2} cos (t) + sin (t) + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.2.3

乘以

cos (t) (- sin (t) cos (t))

$\cos (t) (- \sin (t) \cos (t))$ 。

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解题步骤 3.1.2.3.1

对

cos (t)

$\cos (t)$ 进行

1

$1$ 次方运算。

- {sin (t)}^{2} cos (t) + sin (t) - sin (t) ({cos (t)}^{1} cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) - \sin (t) ({\cos (t)}^{1} \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.2.3.2

对

cos (t)

$\cos (t)$ 进行

1

$1$ 次方运算。

- {sin (t)}^{2} cos (t) + sin (t) - sin (t) ({cos (t)}^{1} {cos (t)}^{1}) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) - \sin (t) ({\cos (t)}^{1} {\cos (t)}^{1}) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.2.3.3

使用幂法则

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ 合并指数。

- {sin (t)}^{2} cos (t) + sin (t) - sin (t) {cos (t)}^{1 + 1} + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) - \sin (t) {\cos (t)}^{1 + 1} + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.2.3.4

将

1

$1$ 和

1

$1$ 相加。

- {sin (t)}^{2} cos (t) + sin (t) - sin (t) {cos (t)}^{2} + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) - \sin (t) {\cos (t)}^{2} + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

- {sin (t)}^{2} cos (t) + sin (t) - sin (t) {cos (t)}^{2} + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) - \sin (t) {\cos (t)}^{2} + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.1.2.4

将

$\cos (t)$ 乘以

$1$ 。

$- {\sin (t)}^{2} \cos (t) + \sin (t) - \sin (t) {\cos (t)}^{2} + \cos (t) + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

解题步骤 3.2

将

$- {\sin (t)}^{2} \cos (t)$ 和

${\sin (t)}^{2} \cos (t)$ 相加。

$\sin (t) - \sin (t) {\cos (t)}^{2} + \cos (t) + \sin (t) {\cos (t)}^{2} + 0$

解题步骤 3.3

将

$\sin (t)$ 和

$0$ 相加。

$- \sin (t) {\cos (t)}^{2} + \cos (t) + \sin (t) {\cos (t)}^{2} + \sin (t)$

解题步骤 3.4

将

$- \sin (t) {\cos (t)}^{2}$ 和

$\sin (t) {\cos (t)}^{2}$ 相加。

$0 + \cos (t) + \sin (t)$

解题步骤 3.5

将

$0$ 和

$\cos (t)$ 相加。

$\cos (t) + \sin (t)$

解题步骤 4

将

$\cos (t) + \sin (t)$ 重写为

$\sin (t) + \cos (t)$ 。

$\sin (t) + \cos (t)$

解题步骤 5

因为两边已证明为相等，所以方程为恒等式。

$\sin^{3} (t) + \cos^{3} (t) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t) = \sin (t) + \cos (t)$ 是一个恒等式

三角学 示例

三角学示例