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The cosecant function, denoted as \( \csc \), is the reciprocal of the sine function. This means that for any angle \( \theta \), the equation \( \csc(\theta) = \frac{1}{\sin(\theta)} \) holds true.
Understanding this reciprocal relationship helps in solving trigonometric equations where sine and cosecant are involved.Key points about the cosecant function:

• It is undefined for angles where sine equals zero. This is because you cannot divide by zero.
• The range of the cosecant function is expressed as \( (-\infty, -1] \cup [1, \infty) \), which means it does not take any values between -1 and 1.
• This function has a periodic cycle of \( 2\pi \), similar to sine, meaning it repeats its values every \( 2\pi \) radians.

When working with cosecant, being able to determine the sine of an angle is crucial as it directly determines the value of its reciprocal function. In the context of the exercise, by finding \( \sin(\frac{7\pi}{4}) = -\frac{\sqrt{2}}{2} \), we could easily compute \( \csc(\frac{7\pi}{4}) = -\sqrt{2} \).

Co-terminal angles are angles that share the same terminal side, essentially differing from each other by full circles, which corresponds to \( 2\pi \) radians or \( 360\) degrees. This means that two angles are co-terminal if their standard angle difference is a multiple of \( 2\pi \).
The main method to find a co-terminal angle is to either add or subtract \( 2\pi \) to the given angle until you reach an angle that falls within the desired interval, usually \([0, 2\pi)\).

• For example, to find a co-terminal angle for \( \frac{15\pi}{4} \), you can subtract \( 2\pi \) to get \( \frac{7\pi}{4} \).
• This helps simplify trigonometric expressions as it allows the angle to be viewed in its simplest form.

In the exercise at hand, the angle \( \frac{15\pi}{4} \) was reduced to \( \frac{7\pi}{4} \) to make the calculations manageable. By using co-terminal angles, complex trigonometric evaluations become simpler, effectively reducing any elongated circle measurements.

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables involved where the functions are defined. These identities are immensely useful for simplifying trigonometric expressions and solving equations.
Some commonly used trigonometric identities include:

• The Pythagorean identities, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• The reciprocal identities: \( \csc(\theta) = \frac{1}{\sin(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• Co-function identities, like \( \sin(\frac{\pi}{2} - \theta) = \cos(\theta) \).

In the context of the exercise, the reciprocal identity was key. By understanding \( \csc(\theta) = \frac{1}{\sin(\theta)} \), one could directly find the inverse cosecant through simplification processes. Knowing these identities allows for more efficient problem-solving and can lead to elegant solutions by substituting equivalent expressions.