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Graph sketching in polar coordinates can be intriguing yet highly rewarding. In polar coordinates, equations are expressed using the polar radius \( r \) and angle \( \theta \). For example, the equation \( r = \cos \theta + \sin \theta \) is such that \( r \) depends on the angular position \( \theta \).
To sketch this graph, start by understanding the relationship between polar and Cartesian systems. Translating polar equations to Cartesian helps visualize them better. Remember that in Cartesian coordinates, \( x = r\cos\theta \) and \( y = r\sin\theta \).
Substituting \( r = \cos \theta + \sin \theta \) into these equations gives,

• \( x = (\cos \theta + \sin \theta)\cos\theta \)
• \( y = (\cos \theta + \sin \theta)\sin\theta \)

Another approach is to visualize or sketch the polar equation directly using software tools designed for graphing trigonometric and polar equations. They can display how changes in \( \theta \) affect \( r \), making the sketching task considerably easier.

Periodic functions, like sine and cosine, repeat their values over a specific interval called the period. For sine and cosine, this period is \( 2\pi \). In our exercise, we encounter a function \( r = \cos \theta + \sin \theta \) which is a sum of the two basic periodic trigonometric functions.
The period of this combined function remains \( 2\pi \) because both \( \cos \theta \) and \( \sin \theta \) complete a full cycle within this interval. Hence, any graph of \( r = \cos \theta + \sin \theta \) will repeat every \( 2\pi \).
This periodic nature is key when determining the full view of the graph. In this exercise, determining the range \( 0 \leq \theta \leq 2\pi \) captures one complete cycle. Understanding periods helps in predicting and sketching the behavior of these periodic functions.

Solving trigonometric equations often uncovers specific angles \( \theta \) where certain conditions hold. For instance, in our equation \( r = \cos \theta + \sin \theta \), we seek when \( r = 0 \). Solving \( \cos \theta + \sin \theta = 0 \) helps identify these critical values of \( \theta \).
Set \( \cos \theta = -\sin \theta \), which simplifies to \( \tan \theta = -1 \). The general solution of \( \tan \theta = -1 \) is \( \theta = \frac{3\pi}{4} + n\pi \), where \( n \) is an integer. Within the interval \( [0, 2\pi] \), the specific solutions are \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{7\pi}{4} \).
Identifying these angles allows us to understand points where the function intersects the pole \( r = 0 \). Mastery of solving trigonometric equations involves knowing their roots and applying transformations like phase shifts or vertical translations to predict graph behavior.