Simple Harmonic Motion: Pendulum theory, trouble understanding

[IQScience]

Homework Statement

So with pendulums in SHM, in my A level physics textbook (AQA Physics A), it shows a pendulum that has been displaced from equilibrium.
It says that the restoring force is provided by the object's weight. Why isn't the restoring force provided by the tension in the string holding the pendulum ball up?
It also says that mgsinθ is the restoring force, because it's the horizontal component of the object's weight, but why does the weight of the object have those components if it's acting straight downwards?

 

[Hertz]

For your first question..

The restoring force is the force that is responsible for there being an equilibrium position. In this sense, you could say that gravity is the restoring force because it is what is responsible for the equilibrium position at \theta = 0. The tension in the string is only responsible for keeping the pendulum a certain distance from the pivot point. If the string was there, but gravity wasn't, the pendulum would still be the same distance from the pivot point, but it wouldn't have a preferred equilibrium position. You see what I'm saying?

Second question..

Let's imagine that the pendulum is hanging downwards (and the weight of the object is downwards as well). If there was no string, the object would fall straight down forever. Its speed would increase, but it would never move side to side.

When there is a wire.. the object would like to fall down, but it can't, because there's a string keeping it the same distance from the pivot point. This means that instead of accelerating downward, it will accelerate along its path, meaning that its acceleration is tangent to its path. If this makes sense then do the following exercise:

Draw a picture of a pendulum hanging down at some random angle theta. (So it's not straight down, it's at an angle theta). Then, draw an arrow downward that represents the force of gravity. Don't worry about the arrow that points towards the pivot point. Now, just looking at the arrow that represents weight, try to make a triangle out of it. In this triangle, the arrow for weight should be the hypotenuse, and there should be a short side of the triangle that is TANGENT to the motion and pointing somewhat downwards. This short side that is tangent to the path is actually the force that will guide the motion of the pendulum.

I'm not sure if this is the exact quantity you're looking for, but it should give you an idea of why sines and cosines show up.