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The cosine function, denoted as \( \cos x \), is one of the basic trigonometric functions. It helps describe the horizontal component of an angle in the unit circle. In a general sense, the value of \( \cos x \) varies between -1 and 1 for all angles in radians. However, within the specific domain of \(0\) to \( \frac{\pi}{2}\), which is the first quadrant of the unit circle, the cosine function value starts at 1 when \( x = 0 \) and decreases to 0 as \( x \) approaches \( \frac{\pi}{2} \). This behavior is crucial because knowing how the cosine function behaves in this domain helps in understanding other trigonometric properties.

Overall, the cosine function plays a vital role in determining angles and distances, particularly when working with right triangles and circular motion.

The secant function, represented as \( \sec x \), is closely related to the cosine function. It is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). This means the secant function is undefined where the cosine function equals zero because division by zero is undefined.

In the domain of \( x \in (0, \frac{\pi}{2}) \), the secant function becomes positive and increases as \( x \) gets closer to \( \frac{\pi}{2} \), moving from 1 to approaching infinity. This is due to \( \cos x \) approaching zero, leading the value of \( \sec x = \frac{1}{\cos x} \) to increase infinitely.

• When \( x = 0 \): \( \sec x = 1 \).
• As \( x \rightarrow \frac{\pi}{2} \): \( \sec x \rightarrow \infty \).

This function is significant in trigonometry, particularly when dealing with secant secures relationships in waveforms and oscillations.

The range of a function refers to the set of all possible output values (\( y \)-values) a function can produce. For the specific function given, \( f(x) = 2 \cos x + \sec^2 x \), determining the range involves considering how each trigonometric component behaves in the defined domain.

From the cosine function, we know within \( x \in (0, \frac{\pi}{2}) \), \( \cos x \) decreases from 1 to 0. Simultaneously, \( \sec^2 x = \left( \frac{1}{\cos x} \right)^2 \) starts at 1 and increases to infinity as \( x \) approaches \( \frac{\pi}{2} \).

• At \( x = 0 \): \( f(x) = 2(1) + 1 = 3 \).
• As \( x \to \frac{\pi}{2} \): \( f(x) \to \infty \).

Hence, within this domain, the function progresses from an output of 3 to values that become infinitely large.
The calculated range for \( f(x) \) is therefore \([3, \infty)\), meaning it includes all real numbers from 3 upwards.