If 3 sin -1 x + cos -1 x = π, then what is x equal to?
Solving Inverse Trigonometric Equations
We are asked to find the value of \(x\) that satisfies the given equation involving inverse trigonometric functions. The equation is:
\(3 \sin^{-1} x + \cos^{-1} x = \pi\)
Using Fundamental Inverse Trigonometric Identities
To solve this equation, we can use a fundamental identity relating \(\sin^{-1} x\) and \(\cos^{-1} x\). The identity is:
\(\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}\), for \(-1 \le x \le 1\)
This identity is crucial for simplifying the given equation.
Simplifying the Equation
Let's rewrite the term \(3 \sin^{-1} x\) in the given equation as \(2 \sin^{-1} x + \sin^{-1} x\). This allows us to group terms to use the identity:
\(2 \sin^{-1} x + \sin^{-1} x + \cos^{-1} x = \pi\)
Now, substitute the identity \(\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}\) into the equation:
\(2 \sin^{-1} x + \frac{\pi}{2} = \pi\)
Solving for sin-1 x
We now have a simpler equation involving only \(\sin^{-1} x\). Let's solve for \(\sin^{-1} x\):
Subtract \(\frac{\pi}{2}\) from both sides of the equation:
\(2 \sin^{-1} x = \pi - \frac{\pi}{2}\)
\(2 \sin^{-1} x = \frac{\pi}{2}\)
Divide both sides by 2:
\(\sin^{-1} x = \frac{\pi/2}{2}\)
\(\sin^{-1} x = \frac{\pi}{4}\)
Finding the Value of x
The equation \(\sin^{-1} x = \frac{\pi}{4}\) means that \(x\) is the number whose sine is \(\frac{\pi}{4}\).
Therefore, we take the sine of both sides:
\(x = \sin\left(\frac{\pi}{4}\right)\)
We know the value of \(\sin\left(\frac{\pi}{4}\right)\) from standard trigonometric values:
\(\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\)
So, the value of \(x\) is:
\(x = \frac{1}{\sqrt{2}}\)
Verifying the Solution
Let's quickly check if \(x = \frac{1}{\sqrt{2}}\) satisfies the original equation \(3 \sin^{-1} x + \cos^{-1} x = \pi\).
If \(x = \frac{1}{\sqrt{2}}\), then \(\sin^{-1} \left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}\) and \(\cos^{-1} \left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}\).
Substitute these values into the left side of the equation:
\(3 \sin^{-1} \left(\frac{1}{\sqrt{2}}\right) + \cos^{-1} \left(\frac{1}{\sqrt{2}}\right) = 3\left(\frac{\pi}{4}\right) + \frac{\pi}{4}\)
\(= \frac{3\pi}{4} + \frac{\pi}{4} = \frac{3\pi + \pi}{4} = \frac{4\pi}{4} = \pi\)
The left side equals the right side (\(\pi\)). Thus, \(x = \frac{1}{\sqrt{2}}\) is the correct solution.
Also, note that \(-1 \le \frac{1}{\sqrt{2}} \le 1\), so the domain condition for \(\sin^{-1} x\) and \(\cos^{-1} x\) is satisfied.
Revision Table: Key Concepts in Inverse Trigonometry
| Concept | Description | Important Identity/Property |
|---|---|---|
| Inverse Sine Function (\(\sin^{-1} x\) or \(\arcsin x\)) | The angle \(\theta\) such that \(\sin \theta = x\), where \(\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) and \(x \in [-1, 1]\). | Domain: \([-1, 1]\) Range: \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) |
| Inverse Cosine Function (\(\cos^{-1} x\) or \(\arccos x\)) | The angle \(\theta\) such that \(\cos \theta = x\), where \(\theta \in [0, \pi]\) and \(x \in [-1, 1]\). | Domain: \([-1, 1]\) Range: \([0, \pi]\) |
| Fundamental Identity | Relationship between inverse sine and cosine for the same argument. | \(\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}\) for \(x \in [-1, 1]\). |
Additional Information: Inverse Trigonometric Function Properties
Understanding the properties of inverse trigonometric functions is essential for solving equations like the one discussed. Here are a few more important points:
- The principal values (ranges) of inverse trigonometric functions are defined to make them one-to-one functions, allowing for a unique inverse.
- Other key identities include:
- \(\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}\) for \(x \in \mathbb{R}\)
- \(\sec^{-1} x + \csc^{-1} x = \frac{\pi}{2}\) for \(x \in (-\infty, -1] \cup [1, \infty)\)
- Remember the domain and range of each inverse function, as these restrict the possible values of \(x\) and the output angle.
- Standard angles (like \(\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\), etc.) and their trigonometric values are frequently used in solving inverse trigonometric problems.
Important Questions from Inverse Trigonometric Functions
What is \(1+\sin ^2\left(\cos ^{-1}\left(\frac{3}{\sqrt{17}}\right)\right)\) equal to ?
What is 2 cot \(\left(\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right)\) equal to ?
Consider the following statements:
1. There exists \({\rm{\theta }} \in \left( { - \frac{{\rm{\pi }}}{2},\frac{{\rm{\pi }}}{2}} \right)\) for which tan -1 (tan θ) ≠ θ
2. \({\sin ^{ - 1}}\left( {\frac{1}{3}} \right) - {\sin ^{ - 1}}\left( {\frac{1}{5}} \right) = {\sin ^{ - 1}}\left( {\frac{{2\sqrt 2 \left( {\sqrt 3 - 1} \right)}}{{15}}} \right)\)
Which of the above statements is/are correct?
Consider the following statements:
1. \({\tan ^{ - 1}}{\rm{x}} + {\tan ^{ - 1}}\left( {\frac{1}{{\rm{x}}}} \right) = {\rm{\pi }}\)
2. There exist x, y ∈ [-1, 1], where x ≠ y such that sin -1 x + cos -1 \({\rm{y}} = \frac{{\rm{\pi }}}{2}\)
Which of the above statements is/are correct?The value of \({\rm{tan}}\left( {2{{\tan }^{ - 1}}\frac{1}{5} - \frac{\pi }{4}} \right)\) is
