91Ó°ÊÓ

To understand parametric equations, think of them as a way to describe a curve using parameters instead of just one coordinate axis. In this exercise, the parameter is represented by the variable \( t \). The parametric equations \( x = \cos 3t \) and \( y = \sin 3t \) describe a curve by defining both the \( x \) and \( y \) coordinates in terms of \( t \).
Using parameters allows us to trace the curve's path based on different values of \( t \).
In this example, as \( t \) ranges from \( 0 \) to \( \pi \), the curve represents part of a circular motion due to the trigonometric functions sine and cosine. This method is particularly useful because it helps visualize curves that may not be easily expressed with a single Cartesian equation.

Derivatives help us understand the rate at which one quantity changes with respect to another. In parametric equations, we often need to find how the \( x \) and \( y \) coordinates change with respect to the parameter \( t \).
Given the equations \( x = \cos 3t \) and \( y = \sin 3t \), we find the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \).
For \( x \), we differentiate: \( \frac{dx}{dt} = -3 \sin 3t \).
For \( y \), the derivative is \( \frac{dy}{dt} = 3 \cos 3t \).
These derivatives reflect how quickly the curve moves in each direction with changes in \( t \), which is essential for calculating arc length.

Integrals allow us to sum up small changes to find total quantities, like the total distance a curve covers, which is known as the arc length.
The formula for arc length of a parametric curve from \( t = a \) to \( t = b \) is:
\[ L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \]
In our example, \( a = 0 \) and \( b = \pi \). By substituting the derivatives, we simplify the integrand using a trigonometric identity (\( \sin^2 3t + \cos^2 3t = 1 \)), which makes the integral more manageable.
This simplification reduces the task to evaluating an easier integral, leading us to find the arc length \( L = 3\pi \).

Trigonometric identities are powerful tools that simplify many calculus problems, including those involving integrals and derivatives.
In this example, we use the identity\( \sin^2 3t + \cos^2 3t = 1 \).
This transforms our formula from a complex expression into a simple constant value, \( \sqrt{9} = 3 \).
Without this identity, evaluating the arc length integral would be much harder. Keeping a good grasp of these identities helps make calculus problems more manageable.

• \( \sin^2\theta + \cos^2\theta = 1 \)
• Useful for simplifying equations involving sinusoidal functions
• Helps in converting products of \( \sin \) and \( \cos \) into a unified form