When we think of electromagnetic waves, we often associate them with radio waves that are picked up by antennas to facilitate communication. These waves exhibit periodic behavior, which can be described using trigonometric functions like sine, cosine, and tangent. In order to fully comprehend wave propagation, it is crucial to also understand the derivatives of these functions.

The derivatives of sine, cosine, and tangent involve even more trigonometric functions, giving us the following expressions:

- derivative of sine: cos x
- derivative of cosine: -sin x
- derivative of tangent: sec^2 x

By applying basic differentiation rules, we can also find the derivatives of other trigonometric functions such as secant, cosecant, and cotangent. Let's explore some examples using sine, cosine, and tangent.

Consider a function involving the sine function. To find its derivative, we can use the derivative of sine, along with the Chain Rule and Power Rule. Starting with a substitution of u = x^2 + 1 and applying the Chain Rule, we get:

**d/dx(sin(x^2+1)) = cos(x^2+1) * (2x)**

Next, we can use the Power Rule to find the derivative of x^2 + 1, which is simply 2x. Substituting this back into the original equation, we can rearrange to find the final derivative.

Now, let's move on to a function involving the cosine function. We can find its derivative using the derivative of cosine, Power Rule, and Chain Rule. Remember that the derivative of cosine is the negative of sine, so for a function involving cos(x^2+1), the derivative would be -sin(x^2+1) * (2x).

Finally, we'll look at a simpler example involving the tangent function. For a function like tan(x^2+1), the derivative would be sec^2(x^2+1) * (2x).

But how do we prove these differentiation rules for trigonometric functions? Let's take a closer look.

In order to derive the derivative of the sine function, we can use the definition of a derivative. By rewriting the expression using the identity for the sine of the sum of two angles, and applying algebraic manipulation, we can simplify the equation. Using limit properties and the Squeeze Theorem, we can then find the values of the involved limits and ultimately solve for the derivative of sine.

Now, let's prove the first limit used in this derivation. By considering a unit circle and examining the triangles shown in the diagram below, we can label each area and use the formulas for area to calculate them. By noting that the base of each triangle is 1 and the height of the isosceles triangle is sin x, we can solve for each area and create an inequality between them. Using the Squeeze Theorem, we can then conclude that the values of sin x approach 0 as x approaches 0.

In conclusion, understanding the derivatives of trigonometric functions is essential for studying wave propagation. By utilizing differentiation rules, as well as proofs based on trigonometric identities, we can fully grasp the behavior and patterns of electromagnetic waves.

In calculus, we often come across the need to find derivatives of trigonometric functions. Here, we will focus on differentiating the sine, cosine, and tangent functions using various methods.

One of the fundamental identities in trigonometry is the Pythagorean trigonometric identity:

**sin ^{2}(x) + cos^{2}(x) = 1**

Using this, we can differentiate both sides of the equation using the chain rule. Since the right-hand side is a constant, its derivative is 0. Therefore, we get:

**2sin(x)cos(x) + (-2)sin(x)cos(x) = 0**

Next, we can substitute the derivative of the sine function, which is the cosine function, and get:

**2sin(x)cos(x) + (-2)cos ^{2}(x) = 0**

Finally, by dividing the equation by 2cos(x) and isolating the derivative of sin(x), we get the result:

**sin'(x) = cos(x)**

We can also use the quotient rule to find the derivative of the tangent function instead of the definition of a derivative. Let's express the tangent function as the quotient of the sine function and cosine function:

**tan(x) = sin(x)/cos(x)**

Now, we can apply the quotient rule, which states that the derivative of a quotient is equal to (numerator x derivative of denominator - denominator x derivative of numerator) divided by (denominator) squared.

Substituting the derivatives of the sine and cosine functions, we get the result:

**tan'(x) = (cos ^{2}(x) + sin^{2}(x)) / cos^{2}(x) = 1/cos^{2}(x) = sec^{2}(x)**

Using the Pythagorean trigonometric identity, we can simplify the numerator to 1, and by recalling that the secant function is the reciprocal of the cosine function, we can further simplify the expression to the final result:

**tan'(x) = sec ^{2}(x)**

Ultimately, using the quotient rule is a faster and easier method compared to using the definition of a derivative in this case.

From the above, we can conclude three key takeaways about differentiating the sine, cosine, and tangent functions:

- The derivative of the sine function is the cosine function, i.e.
**sin'(x) = cos(x)**. - The derivative of the cosine function is the negative of the sine function, i.e.
**cos'(x) = -sin(x)**. - The derivative of the tangent function is the secant function squared, i.e.
**tan'(x) = sec**.^{2}(x)

For proving the derivatives of the sine and cosine functions, we use two important limits: **lim(x -> 0) sin(x)/x = 1** and **lim(x -> 0) (1 - cos(x))/x = 0**. And finally, the derivative of the tangent function can be found using either the quotient rule or the definition of a derivative.

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