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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.

We know,

To solve the equation tan⁡−1(2x)+tan⁡−1(3x)=π4tan−1(2x)+tan−1(3x)=4π, we'll use the tangent addition formula:

tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡α⋅tan⁡βtan(α+β)=1−tanα⋅tanβtanα+tanβ

Let α=tan⁡−1(2x)α=tan−1(2x) and β=tan⁡−1(3x)β=tan−1(3x). Then, we have:

tan⁡(α+β)=2x+3x1−2x⋅3xtan(α+β)=1−2x⋅3x2x+3x

tan⁡(α+β)=5x1−6x2tan(α+β)=1−6x25x

Given that tan⁡(α+β)=π4tan(α+β)=4π, we have:

5x1−6x2=π41−6x25x=4π

Cross-multiply:

5x⋅4=π(1−6x2)5x⋅4=π(1−6x2)

20x=π−6πx220x=π−6πx2

6πx2+20x−π=06πx2+20x−π=0

Now, solve this quadratic equation for xx. We can use the quadratic formula:

x=−b±b2−4ac2ax=2a−b±b2−4ac

where a=6πa=6π, b=20b=20, and c=−πc=−π:

x=−20±(20)2−4⋅6π⋅(−π)2⋅6πx=2⋅6π−20±(20)2−4⋅6π⋅(−π)

x=−20±400+24π212πx=12π−20±400+24π2

x=−20±400+24π212πx=12π−20±400+24π2

So, the solutions for xx are:

x=−20+400+24π212πx=12π−20+400+24π2

x=−20−400+24π212πx=12π−20−400+24π2

These are the solutions for the given equation.

In LaTeX code:

x=−20+400+24π212πx=12π−20+400+24π2

x=−20−400+24π212πx=12π−20−400+24π2

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