Geometry 几何

theorems 定律

1)In any triangle $ABC$, $AB\lt AC+BC$.
1)对于任意三角形 $ABC$$AB\lt AC+BC$
2)For any right-angled triangle $ABC$ with side $a$ as hypotenuse, $a^2=b^2+c^2$.
2)对于任意以 $a$ 为斜边的直角三角形 $ABC$$a^2=b^2+c^2$
3)For any triangle $ABC$, $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$.
3)对于任意三角形 $ABC$$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$
4)For any triangle $ABC$, $a^2=b^2+c^2-2bc\cos A$.
4)对于任意三角形 $ABC$$a^2=b^2+c^2-2bc\cos A$

proof 证明

Please try to prove 2) and 3) yourself using size relationships.

Trigonometry 三角学

theorems 定律

1) $\sin ^2 \theta + \cos ^2 \theta = 1 {\kern 5em} ( \sin ^2 \theta = ( \sin \theta )^2)$
2) $\sin (A\pm B) =\sin A \cos B \pm \cos A\sin B$
3) $\cos (A\pm B)=\cos A\cos B \mp \sin A\sin B$
4) $\tan (A\pm B)=\dfrac{\tan A \pm \tan B}{1\mp \tan A\tan B}$
5) $\sin 2\alpha =2\sin \alpha \cos \alpha$
6) $\cos 2\alpha = \cos ^2\alpha -\sin^2\alpha =2\cos ^2\alpha -1=1-2\sin ^2 \alpha$
7) $\tan 2\alpha = \dfrac{2\tan \alpha}{1-\tan ^2\alpha}$
8) $\sin ^2 A=\dfrac{1-\cos 2A}{2}$
9) $\cos ^2 A=\dfrac{1+\cos 2A}{2}$
10) $\tan ^2 A=\dfrac{1-\cos 2A}{1+\cos 2A}$
11) For a unit circle on a coordinate grid with its center at the origin, point $P(\cos \theta ,\sin \theta )$ is on the circle, and line $\overline{OP}$ and the x-axis form an angle of $\theta$.
11)对于一个表示在坐标轴上且圆心在原点的单位圆，点 $P(\cos \theta ,\sin \theta )$ 一定在该圆上，而且直线 $\overline{OP}$ 和x-轴的夹角为 $\theta$
12) $\tan (90^\circ -A)=\cot (A)$

proof 证明

11) will be used but not proved 11) 会被用来证明，但它本身不会被证明
Try to prove 1),5),6),7).

Other proofs 其他证明:

Note by Jeff Giff
10 months, 2 weeks ago

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@Páll Márton, I’ve been thinking of a proof for 8,9,10, but I have no idea. Can you help me?

- 10 months, 2 weeks ago

- 10 months, 2 weeks ago

Never mind :) I finally found the proof online

- 10 months, 2 weeks ago

LOL Your first(second) line is wrong

- 10 months, 2 weeks ago

oops... third

- 10 months, 2 weeks ago

or fourth with the title

- 10 months, 2 weeks ago

Where?

- 10 months, 2 weeks ago

Wait...whoa! Thanks!

- 10 months, 2 weeks ago

ABC, AB\gt AC+BC

- 10 months, 2 weeks ago

P.S. I added one proof to the chain

- 10 months, 2 weeks ago

In China the correct order is 2,3,1,5,6,7??? lol

- 10 months, 2 weeks ago

Umm... actually there are two parts, and the numbers refer to the theorem in the part :)

- 10 months, 2 weeks ago

LOL

- 10 months, 2 weeks ago