Problem 207 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } 4 \sin ^{-1} x+\cos ^{-1} x=\pi, \text { then find } x \text { . } $$

Step-by-Step Solution

Answer

The value for \(x\) is \(x = \cos(\frac{\pi}{3})\) which simplifies to \(x = \frac{1}{2}\)

Step 1: Apply trigonometric identity

First, apply the standard identity \(\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}\) to simplify the equation. Multiply both sides of the identity equation by 4 to get \(4 \sin^{-1}x + 4\cos^{-1}x = 2\pi\). This modification allows us to replace \(4 \sin^{-1}x + 4\cos^{-1}x\) term.

Step 2: Substitute in the provided equation

Now substitute \(4 \sin^{-1}x + 4\cos^{-1}x = 2\pi\) in the original equation \(4 \sin^{-1} x + \cos^{-1} x = \pi\). This substitution will give us \(2\pi - 3\cos^{-1}x = \pi\). So, this effectively simplifies our equation to \(-3\cos^{-1}x = -\pi\).

Step 3: Solve for x

Finally, we isolate \(x\) from the equation \(-3\cos^{-1}x = -\pi\) by dividing both sides by -3. This gives us \(\cos^{-1}x = \frac{\pi}{3}\). Since \(\cos^{-1}\) is inverse operation of cosine, the solution for \(x\) can be found by taking the cosine of \(\pi/3\). Thus, \(x = \cos(\frac{\pi}{3})\).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equalities that involve trigonometric functions. They are extremely useful tools for solving a wide range of problems in mathematics and science. One such important identity is \( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \). This identity reflects the complementary nature of sine and cosine for a given angle in a right triangle. By understanding and applying these identities, we can simplify complex trigonometric equations and make them more manageable. In the given problem, we multiply both sides of the trigonometric identity by 4 to relate it to the equation given in the exercise. This alteration helps us strategically substitute and simplify the given equation, aiding in the solution process.

Solving Trigonometric Equations

To solve trigonometric equations, one needs to identify relationships or identities that can aid in simplifying the equation. It often involves employing trigonometric identities, isolating the variable, and using inverse trigonometric functions to find the solution.

In the problem provided, we start with the expression \(4 \sin^{-1}x + \cos^{-1}x = \pi\). By utilizing the identity \( \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \), we are able to reframe the equation by substituting the equivalent expression into the given problem.

• We substitute \(4 \sin^{-1}x + 4\cos^{-1}x = 2\pi\) into the equation, resulting in \(2\pi - 3\cos^{-1}x = \pi\).
• This equivalence simplifies to \(-3\cos^{-1}x = -\pi\).
• Finally, by isolating \(\cos^{-1}x\), we divide both sides by -3, which leads us to the resolution of the problem.

These steps demonstrate how breaking down and strategically manipulating the components of a trigonometric equation can efficiently lead to a solution.

Arcsine Function

The arcsine function, denoted as \( \sin^{-1} \) is an inverse trigonometric function that takes on the range of the angles measured between \(-\frac{\pi}{2}\) to \( \frac{\pi}{2} \). When applied to a value, it reveals the angle whose sine is that value.

This function is crucial in scenarios where we know the sine of an angle but need to find the actual angle. The arcsine, along with other inverse trigonometric functions like arccosine (\( \cos^{-1} \)) and arctangent, are tools used to resolve equations where the angle is the unknown.

In this exercise, identifying the relationship involving the arcsine and arccosine functions allows us to express the given equation in a solvable manner through the application of trigonometric identities. These relationships and the familiarity with the inverse trigonometric functions greatly simplify the problem-solving process.