Need help understanding finding this force using trig

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The discussion revolves around understanding the forces acting on a block resting on an inclined plane, specifically how to correctly apply trigonometry to resolve these forces. The block has a mass of 40kg, and the user is confused about calculating the normal force (N) and its relationship with the weight (w) of the block. The correct calculation shows that N is derived from the weight multiplied by the cosine of the angle of inclination, but the user mistakenly attempts to find the hypotenuse instead of recognizing that the weight is the hypotenuse of the triangle formed by the components. This misunderstanding is common when resolving vectors into components, emphasizing the importance of correctly identifying the right triangle in such problems. Clarifying these concepts will aid in solving similar physics problems in the future.
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Homework Statement


The block resting on the inclined plane shown has a mass of 40kg. Determine the maximum and minimum value of P for which the block is in equilibrium. (fs=0.35 and θ=25°)

The image on top is the diagram in the book and the image below it is my free-body diagram (not too sure if the Fr (friction) force is correct though)

prob_zpsu89wywac.png


3. Attempt at the solution
Now my problem isn't exactly but how to solve this, since I found the solution online. My question is more about the trigonometry used to find some of the forces in this diagram. I found that the w (the weight) is of course (40)(9.8) = 392.4. Now the question is this:

The solution and the book claims that N=(392.4)(cos 25) = 355.64. However I am confused because if I say that:
cos 25 = a/h (N)
cos 25 = 392.4/h
h=392.4/cos 25 = 432.97
Which is incorrect. Can some explain why in my mind, I am 100% that this equation does indeed solve for the hypotenuse, yet the book claims that it is the product of the two? Is my free-body diagram wrong? I am certain i that it isn't (except maybe the friction force which may be wrong). Once I understand this I am sure I can better understand the solution to this problem and other similar problems.
 
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You are using the wrong triangle to calculate the components of the weight. (A very common error, by the way, so don't feel bad.) Note that when you resolve a vector (in this case the weight) into components using a right triangle, the full vector is always the hypotenuse of that triangle.

Read this: Inclined Planes
 
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