The Fundamental Theorem of Calculus (FTC) is an important concept that connects two main ideas in math: differentiation and integration. This is especially useful when we work with functions like sine and cosine. Let's go through it step by step.

Understanding the Fundamental Theorem of Calculus

The FTC has two main parts:

1. Part 1: If a function ( f ) is continuous on the interval from ( a ) to ( b ), then we can create another function ( F ) like this:

   $$F(x) = \int_a^x f(t) \, dt$$

   This new function ( F ) will also be continuous on that interval, it can be differentiated between ( a ) and ( b ), and the derivative ( F' ) will equal ( f(x) ).

2. Part 2: If ( f ) is continuous on the same interval, we have:

   $$\int_a^b f(x) \, dx = F(b) - F(a)$$

   Here, ( F ) is any antiderivative of ( f ).

Applying the Theorem to Trigonometric Functions

Trigonometric functions, like ( \sin(x) ) and ( \cos(x) ), are continuous and differentiable everywhere. This makes them great examples for using the FTC.

Example 1: Integrating ( \sin(x) )

Let’s look at how to find the integral of ( \sin(x) ) from ( 0 ) to ( \pi ):

1. Find the Antiderivative: We know that the antiderivative of ( \sin(x) ) is ( -\cos(x) ). So we write:

   $$F(x) = -\cos(x).$$

2. Evaluate the Definite Integral: Now, we can use Part 2 of the FTC:

   $$\int_0^\pi \sin(x) \, dx = F(\pi) - F(0) = [-\cos(\pi)] - [-\cos(0)].$$

   Plugging in the values gives us:

   $$= [1] - [-1] = 1 + 1 = 2.$$

So, the area under the curve of ( \sin(x) ) from ( 0 ) to ( \pi ) is ( 2 ).

Example 2: Integrating ( \cos(x) )

Now, let's try the function ( \cos(x) ) over the interval from ( 0 ) to ( \frac{\pi}{2} ):

1. Find the Antiderivative: The antiderivative of ( \cos(x) ) is ( \sin(x) ), so we write:

   $$F(x) = \sin(x).$$

2. Evaluate the Definite Integral: Again, we will use the FTC:

   $$\int_0^{\frac{\pi}{2}} \cos(x) \, dx = F\left(\frac{\pi}{2}\right) - F(0) = [\sin\left(\frac{\pi}{2}\right)] - [\sin(0)].$$

   This simplifies to:

   $$= 1 - 0 = 1.$$

So, the area under the curve of ( \cos(x) ) from ( 0 ) to ( \frac{\pi}{2} ) is ( 1 ).

Summary

In short, using the Fundamental Theorem of Calculus with trigonometric functions is pretty simple since they’re continuous and can be easily differentiated. By finding their antiderivatives and plugging in our limits, we can figure out definite integrals easily.

This approach works for all trigonometric functions, making the FTC a powerful tool in calculus.

Remember, integrating trigonometric functions not only gives you the areas under curves but also helps you understand the important connection between integration and differentiation, which builds a strong foundation for more advanced studies in calculus!