Why Cos Theta Is Used In Formula Of Work Done ?

Why cosine is used in formula of work done?

We know that the formula of the work done is product of Force and Displacement i.e force x displacement. For more precise we should present the formula as, work done by the force multiplied  by the displacement in the direction of the force. So,  we use cosine in the formula of work done. First let us know;

What is the work done and its SI unit? 

In physics, a work is done if a force makes a movement of an object. It is also known as the transfer of energy. The simple definition of the work done is, 'It is the product of force and displacement. Its SI unit is Joule (J). 
The formula to calculate the work done is: 
Work Done = Force * Displacement. 
or, W = F x s

Why is Cosθ used to calculate Formula of  work done? 

Cosine in formula of work done

With the help of above picture we can understand the use of cos theta (cosθ) in the work equation. If an object moves along X-axis (horizontal surface) and resultant force makes an angle (θ) with horizontal, then we use cos theta (cosθ)  in formula of the work done. So, the work is cosine. 
Cosθ = base(b)/hypotenuse(h). 
The component of x-axis will become Fcos(theta). 
Hence W=fxs becomes W=s x f x cosθ.

For Simplicity, 
Formula of work done

Let us suppose, an object is rolled in the floor(along the x-axis). It does not make an angle with the x-axis. i.e. the object makes zero degree with floor. then, 
W= F x s x cosθ 
or, W= F x s x cos0°
or, W = F x s x 1 [Since cos0°=1] 
Hence, W = F x s. 
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Conclusion: If the resultant force makes angle zero degree with horizontal, then it is not necessary to use cos theta (cosθ) in the formula of work done. However, we must use cosine in the work done if the resulatant force makes certain angle with x-axis.

When to use sine in the formula of the work? 

We can use sine in the formula of work done, if work is done against the gravity along y-axis (vertical direction). If  phi (Φ) is the angle in direction of y-axis, then the formula of work done will be, 
W=F x s x SinΦ 
or, W = f x s x SinΦ

Derivation of Work formula: 

From the defination of work done, 
W = F x s x cosθ
W=work done 
s=displacement of an object
 F=force acting on an object 
Cos(theta)=angle admist force and displacement.

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