In daily life, we often encounter the problem of finding the "area", such as: the area of a triangle is equal to: "1/2*base*height", the area of parallelogram is equal to: "base*height", etc. . Sometimes, "area" can solve other mathematical "difficulties"!
Look at an example. There is a triangle ABC in the coordinate paper . The length of the coordinate unit is 1. What is the value of sin∠A?
Analyze this problem: first we have to find sin∠A in the "right triangle", so we can use C to make the "height" on the AB side and set the height to h. How to find the height? It seems that we can only find the "area" of triangle ABC. We found that triangle ABC is in one: "3*5 rectangle", and the area of triangle ABC can be obtained by using the "division method". This problem is solved.
The first step is to find the area "S" of the triangle ABC, which is equal to the area of the "outer rectangle" minus the area of the "three" right triangles, namely: S=3*5 -1/2*(3*1+5*2+4*1)=15-17/2=13/2.
The second step, to find the length of the AB side, use Pythagorean theorem : AB*AB=5*5+2*2=29, so: AB=, the same: AC=.
In the third step, use the area formula : S=1/2*AB* h=1/2**h=13/2 to solve: h=13/29.
The fourth step is to use the trigonometric function to define: In a right triangle,The sine function is equal to: "opposite side than hypotenuse", namely: sin∠A=h/AC=(13/29)/=13/493.
It turns out that the "area" is so powerful, take the "area" to check in, and use the "area" to speak!
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