Mathematics in senior three, why the sine value of dihedral angle is equal to the absolute value of two normal vector cosine values, why not draw a picture to explain, thank you.

As shown in the figure, A is a dihedral angle, two sides of A are two planes, and two sides of B are normal vectors. Therefore, the dihedral angle and included angle between two normal vectors are either equal or complementary (the direction of view vector selection), but in general, their cosine values are equal in absolute value, and the sine value of dihedral angle is not equal to the cosine values of normal vectors on both sides (except 45 degrees). Look at the topic, people calculate the cosine value first, and then the sine value (it says "so" here), not that the cosine value is equal to the sine value (you asked why these two values are equal, but the topic didn't explain this, but you misunderstood it).

Another topic, sine equals cosine, refers to the angle between a straight line and a plane, not a dihedral angle. The two are not the same thing. The normal vector of a line and a plane, and the projection of the normal vector (see the figure below, ABC, ACB are the included angle between the line and the plane, and BAC is the included angle between the normal vector and the line) form a right triangle, so A+C=90 degrees, which is why you said "sine value equals cosine value". Again, here refers to straight lines and planes, not dihedral angles. These conclusions are not rote learning, let alone all situations. Be sure to find this relationship in specific topics.

Look at examples and learn ideas. What clues are used to solve the problem? Why is this an example? If not, is there any other way? Instead of trying to understand this problem, you don't understand this, but you are obsessed with local results. It's no use looking at more examples.