How Do You Calculate the Angle and Area in Trigonometry Problems?

[rocomath]

Homework Statement

32. Three gears are arranged as shown in the figure below. Find the angle $$\phi$$

46. Prove that the area of a parallelogram is the product of two adjacent sides and the sine of the included angle.

Homework Equations

The Attempt at a Solution

32. I assumed that since the gears touch that I could add their radius. Would that be incorrect? I was thinking I had to use the linear and angular speed formulas, but I wasn't sure. I know that their linear speed is equal, but I don't think I need to do all that.

46. I followed the steps in which they obtained the Area of a triangle.

 

[Gib Z]

Both workings seem find to me.

 

[Architeuthis Dux]

I used the formula for the area of a triangle, which is 1/2 base x height, and then substituted the base with one of the adjacent sides and the height with the sine of the included angle. This resulted in the formula for the area of a parallelogram: A = a * b * sin(theta). To prove this, I used the fact that the area of a parallelogram can also be calculated by taking the cross product of two adjacent sides (a and b) and the sine of the included angle (theta). This is because the cross product of two vectors gives the area of the parallelogram formed by those vectors. Therefore, the area of the parallelogram can be written as A = |a x b| = |a| * |b| * sin(theta) = a * b * sin(theta). This shows that the area of a parallelogram is indeed the product of two adjacent sides and the sine of the included angle.