三角形计算器

计算器将尝试找出三角形的所有边和角(直角三角形、钝角三角形、锐角三角形、等腰三角形、等边三角形),以及它的周长和面积,并显示步骤。

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您的输入

求解三角形,如果$$$a = 9$$$, $$$b = 10$$$, $$$C = 45^0$$$

解决方案

根据余弦定律: $$$c^{2} = a^{2} + b^{2} - 2 a b \cos{\left(C \right)}$$$

在我们的例子中, $$$c^{2} = 9^{2} + 10^{2} - \left(2\right)\cdot \left(9\right)\cdot \left(10\right)\cdot \left(\cos{\left(45^0 \right)}\right) = 181 - 90 \sqrt{2}$$$

因此, $$$c = \sqrt{181 - 90 \sqrt{2}}$$$

根据正弦定律: $$$\frac{a}{\sin{\left(A \right)}} = \frac{c}{\sin{\left(C \right)}}$$$

在我们的例子中, $$$\frac{9}{\sin{\left(A \right)}} = \frac{\sqrt{181 - 90 \sqrt{2}}}{\sin{\left(45^0 \right)}}$$$

因此, $$$\sin{\left(A \right)} = \frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}}$$$

有两种可能的情况:

  1. $$$A = \left(\frac{180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)}}{\pi}\right)^0$$$

    第三个角是$$$B = 180^0 - \left(A + C\right)$$$

    在我们的例子中, $$$B = 180^0 - \left(\left(\frac{180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)}}{\pi}\right)^0 + 45^0\right) = \left(\frac{- \pi \left(45 + \frac{180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)}}{\pi}\right) + 180 \pi}{\pi}\right)^0$$$

    面积为$$$S = \frac{1}{2} a b \sin{\left(C \right)} = \left(\frac{1}{2}\right)\cdot \left(9\right)\cdot \left(10\right)\cdot \left(\sin{\left(45^0 \right)}\right) = \frac{45 \sqrt{2}}{2}$$$

    周长是$$$P = a + b + c = 9 + 10 + \sqrt{181 - 90 \sqrt{2}} = \sqrt{181 - 90 \sqrt{2}} + 19$$$

  2. $$$A = \left(\frac{- 180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)} + 180 \pi}{\pi}\right)^0$$$

    第三个角是$$$B = 180^0 - \left(A + C\right)$$$

    在我们的例子中, $$$B = 180^0 - \left(\left(\frac{- 180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)} + 180 \pi}{\pi}\right)^0 + 45^0\right) = \left(\frac{- \pi \left(45 + \frac{- 180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)} + 180 \pi}{\pi}\right) + 180 \pi}{\pi}\right)^0$$$

    这种情况是不可能的,因为与较长边相对的角度必须更大。

回答

$$$a = 9$$$A

$$$b = 10$$$A

$$$c = \sqrt{181 - 90 \sqrt{2}}\approx 7.329446049083208$$$A

$$$A = \left(\frac{180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)}}{\pi}\right)^0\approx 60.258581489369345^0$$$A

$$$B = \left(\frac{- \pi \left(45 + \frac{180 \operatorname{asin}{\left(\frac{9 \sqrt{2}}{2 \sqrt{181 - 90 \sqrt{2}}} \right)}}{\pi}\right) + 180 \pi}{\pi}\right)^0\approx 74.741418510630655^0$$$A

$$$C = 45^0$$$A

面积: $$$S = \frac{45 \sqrt{2}}{2}\approx 31.819805153394639$$$A

周长: $$$P = \sqrt{181 - 90 \sqrt{2}} + 19\approx 26.329446049083208$$$A