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The sine function, denoted as \( \sin x \), is one of the fundamental trigonometric functions. It is based on the unit circle, which is a circle with a radius of one. In a right-angled triangle, sine of an angle \( x \) is defined as the ratio of the length of the side opposite the angle to the hypotenuse (the longest side of the triangle). This can be expressed as:

• \( \sin x = \frac{\text{opposite}}{\text{hypotenuse}} \)

When dealing with the unit circle, the sine of angle \( x \) is simply the y-coordinate of the point where the angle intersects with the circle.
The sine function is periodic with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians. It varies between -1 and 1, reaching its maximum at \( x = \frac{\pi}{2} \), and its minimum at \( x = \frac{3\pi}{2} \). This oscillating characteristic is what gives the sine function its wave-like shape.

The secant function, written as \( \sec x \), is the reciprocal of the cosine function. Hence, it is defined as:

• \( \sec x = \frac{1}{\cos x} \)

The cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the hypotenuse. Thus, secant relates to the hypotenuse over the adjacent side.
In terms of the unit circle, where the hypotenuse is always 1, the secant function is simply the reciprocal of the x-coordinate of the point intersecting the circle.
Because it involves division by the cosine function, \( \sec x \) is undefined when \( \cos x = 0 \), specifically at angles like \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \). Like all trigonometric functions, secant is periodic, with a period of \( 2\pi \). Its graph displays vertical asymptotes wherever cosine is zero, and it has no maximum or minimum since it can approach infinity.

The tangent function, indicated as \( \tan x \), is the ratio of two other primary trigonometric functions: sine and cosine. It is defined as:

• \( \tan x = \frac{\sin x}{\cos x} \)

In a right-angled triangle, it is the ratio of the length of the opposite side to the length of the adjacent side. In unit circle terms, it represents the slope of the line connecting the origin with the point on the circumference.
Tangent is periodic with a period of \( \pi \) because its values repeat every \( \pi \). This is due to the fact that it effectively wraps around the circle twice as fast as the sine and cosine functions, which have a \( 2\pi \) period.
Meanwhile, \( \tan x \) is undefined where \( \cos x = 0 \), at points like \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \), where its graph exhibits vertical asymptotes. At these points, tangent's value tends towards infinity. Its unique property of oscillating between positive and negative infinity within its period gives it a distinct repetitive slope pattern.