Question with arcsin and arccos

  • MHB
  • Thread starter Amer
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Pythagorean identity, we can simplify the equation to:$$\sqrt{x}\left(\sqrt{1-x^2}+\sqrt{x-1}\right)=0$$In summary, to find x such that $\sin^{-1}(x)+\cos^{-1}\left(\frac{1}{\sqrt{x}}\right)=0$, we can use the Pythagorean identity to simplify the equation to $\sqrt{x}\left(\sqrt{1-x^2}+\sqrt{x-1}\right)=0$. This leads to solving an irrational equation.
  • #1
Amer
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Find x such that

[tex]sin^{-1} (x) + cos^{-1}\left( \frac{1}{\sqrt{x}}\right) = 0 [/tex]
 
Last edited:
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  • #2
Amer said:
Find x such that

[tex]sin^{-1} (x) + cos^{-1}\left( \frac{1}{\sqrt{x}}\right) = 0 [/tex]

First let's take the $\sin$ of both sides:
$$\sin\left[\mbox{arc}\sin(x)+\mbox{arc}\cos\left(\frac{1}{ \sqrt{x}}\right)\right]=\sin(0)$$
$$\Leftrightarrow \sin\left[\mbox{arc}\sin(x)\right]\cos\left[\mbox{arc}\cos\left(\frac{1}{ \sqrt{x}}\right)\right]+\sin\left[\mbox{arc}\cos\left(\frac{1}{ \sqrt{x}}\right)\right]\cos\left[\mbox{arc}\sin(x)\right]=0$$
$$\Leftrightarrow x\left(\frac{1}{\sqrt{x}}\right)+\sqrt{1-\left(\frac{1}{\sqrt{x}}\right)^2}\sqrt{1-x^2}=0$$
$$\Leftrightarrow \sqrt{x}+\sqrt{1-\frac{1}{x}}\sqrt{1-x^2}=0$$
$$\Leftrightarrow \sqrt{x}+\sqrt{\frac{x-1}{x}}\sqrt{1-x^2}=0$$
$$\Leftrightarrow \ldots$$

Now you have to solve an irrational equation.
 
Last edited:
  • #3
Thanks
 

Related to Question with arcsin and arccos

1. What is the difference between arcsin and arccos?

Both arcsin and arccos are inverse trigonometric functions, but they have different outputs. Arcsin gives the measure of an angle in radians whose sine is a given number, while arccos gives the measure of an angle in radians whose cosine is a given number. In other words, arcsin gives the angle whose sine is a given number, and arccos gives the angle whose cosine is a given number.

2. What is the domain and range of arcsin and arccos?

The domain of both arcsin and arccos is -1 to 1, as the outputs of sine and cosine are limited to this range. The range of arcsin is -π/2 to π/2, while the range of arccos is 0 to π.

3. How do you solve equations involving arcsin and arccos?

To solve equations involving arcsin and arccos, you can use algebraic manipulation and trigonometric identities. It is also important to remember the domain and range of these functions when solving equations, as their outputs are limited to a specific range.

4. Can you use arcsin and arccos on a calculator?

Yes, most calculators have buttons for arcsin and arccos, usually labeled as "sin-1" and "cos-1" respectively. Just make sure to check that your calculator is set to the correct angle mode (degrees or radians) before using these functions.

5. What are some real-life applications of arcsin and arccos?

Arcsin and arccos are commonly used in fields such as physics, engineering, and navigation to calculate angles and distances. They are also used in digital signal processing, statistics, and in the study of harmonic motion.

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