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A relation R:A→AR:A→A is said to be symmetric if for every pair of elements a,ba,b in set AA, whenever (a,b)(a,b) is in RR, then (b,a)(b,a) must also be in RR. In other words, if aa is related to bb, then bb must be related to aa as well, for all a,ba,b in AA.

A universal relation in the context of relational databases refers to a relation (or table) that contains all possible combinations of tuples from the sets involved. In simpler terms, it includes every possible pair of elements from its constituent sets.

For example, let's consider a universal relation that represents the Cartesian product of the sets A = {1, 2} and B = {x, y}. The universal relation would contain all possible combinations of elements from A and B:

In this example, the universal relation contains all possible combinations of elements from set A and set B.

Inverse trigonometric functions are functions that "undo" the effects of trigonometric functions. They provide a way to find the angle (or value) associated with a given trigonometric ratio. Inverse trigonometric functions are denoted by \(\sin^{-1}(x)\), \(\cos^{-1}(x)\), \(\tan^{-1}(x)\), \(\cot^{-1}(x)\), \(\sec^{-1}(x)\), and \(\csc^{-1}(x)\), representing arcsine, arccosine, arctangent, arccotangent, arcsecant, and arccosecant, respectively.

Here's a brief explanation of each inverse trigonometric function:

1. **arcsin (or \(\sin^{-1}(x)\))**: Gives the angle whose sine is \(x\), where \(x\) is between -1 and 1.

2. **arccos (or \(\cos^{-1}(x)\))**: Gives the angle whose cosine is \(x\), where \(x\) is between -1 and 1.

3. **arctan (or \(\tan^{-1}(x)\))**: Gives the angle whose tangent is \(x\).

4. **arccot (or \(\cot^{-1}(x)\))**: Gives the angle whose cotangent is \(x\).

5. **arcsec (or \(\sec^{-1}(x)\))**: Gives the angle whose secant is \(x\), where \(x \geq 1\) or \(x \leq -1\).

6. **arccsc (or \(\csc^{-1}(x)\))**: Gives the angle whose cosecant is \(x\), where \(x \geq 1\) or \(x \leq -1\).

It's important to note that the range of inverse trigonometric functions is restricted to ensure that they are single-valued and have unique inverses. The specific range depends on the convention used, but commonly accepted ranges are as follows:

- For arcsin and arccos: \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\) (or \(-90^\circ \leq \theta \leq 90^\circ\)).
- For arctan: \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\) (or \(-90^\circ < \theta < 90^\circ\)).
- For arccot: \(0 < \theta < \pi\) (or \(0^\circ < \theta < 180^\circ\)).
- For arcsec and arccsc: \(0 \leq \theta < \frac{\pi}{2}\) and \(\frac{\pi}{2} < \theta \leq \pi\) (or \(0^\circ \leq \theta < 90^\circ\) and \(90^\circ < \theta \leq 180^\circ\)).

These functions are essential in solving trigonometric equations, modeling periodic phenomena, and various applications in science, engineering, and mathematics.

To find the principal value of sin⁡−1(−3−2)sin⁡−1(−32)sin−1(−3−2)sin−1(−23), we'll start by finding the individual principal values of sin⁡−1(−3−2)sin−1(−3−2

) and sin⁡−1(−32)sin−1(−23), and then multiply them together.

1. sin⁡−1(−3−2)sin−1(−3−2

1. ):

Since sin⁡−1sin−1 gives an angle whose sine is equal to the given value, we need to find an angle θθ such that sin⁡θ=−3−2sinθ=−3−2

.

However, the sine function only returns values between -1 and 1. Therefore, −3−2−3−2

is outside the range of the sine function, and there is no real angle θθ for which sin⁡θ=−3−2sinθ=−3−2. Hence, sin⁡−1(−3−2)sin−1(−3−2

) is undefined.

2. sin⁡−1(−32)sin−1(−23):

Again, since sin⁡−1sin−1 gives an angle whose sine is equal to the given value, we need to find an angle ϕϕ such that sin⁡ϕ=−32sinϕ=−23.

Similar to the previous case, −32−23 is outside the range of the sine function, and there is no real angle ϕϕ for which sin⁡ϕ=−32sinϕ=−23. Hence, sin⁡−1(−32)sin−1(−23) is also undefined.

Therefore, the principal value of sin⁡−1(−3−2)sin⁡−1(−32)sin−1(−3−2

To find the principal value of cos⁡−1(3−2)cos⁡−1(−32)cos−1(3−2)cos−1(−23), we'll start by finding the individual principal values of cos⁡−1(3−2)cos−1(3−2

) and cos⁡−1(−32)cos−1(−23), and then multiply them together.

1. cos⁡−1(3−2)cos−1(3−2

1. ):

Since cos⁡−1cos−1 gives an angle whose cosine is equal to the given value, we need to find an angle θθ such that cos⁡θ=3−2cosθ=3−2

.

However, the cosine function only returns values between -1 and 1. Therefore, 3−23−2

is outside the range of the cosine function, and there is no real angle θθ for which cos⁡θ=3−2cosθ=3−2. Hence, cos⁡−1(3−2)cos−1(3−2

) is undefined.

2. cos⁡−1(−32)cos−1(−23):

Similarly, −32−23 is outside the range of the cosine function, and there is no real angle ϕϕ for which cos⁡ϕ=−32cosϕ=−23. Hence, cos⁡−1(−32)cos−1(−23) is also undefined.

Therefore, the principal value of cos⁡−1(3−2)cos⁡−1(−32)cos−1(3−2