Question

If \(\sin ^{-1}\left(\frac{\mathrm{x}}{5}\right)+\operatorname{cosec}^{-1}\left(\frac{5}{4}\right)=\frac{\pi}{2}, \mathrm{x}\) equals (a) 3 (b) 4 (c) 5 (d) 6

Short answer

#Short Answer# The answer to the given equation given the constraints is \(x = 4\), which corresponds to choice (b).

Step-by-step solution

Use the identity that relates \(\sin^{-1}\) and \(\operatorname{cosec}^{-1}\)

We have the following identity for arc functions involving sine and cosecant: \(\sin^{-1}(a) + \operatorname{cosec}^{-1}(a) = \frac{\pi}{2}\) Therefore we can write our equation as following: \(\sin^{-1}\left(\frac{x}{5}\right) + \operatorname{cosec}^{-1}\left(\frac{5}{4}\right) = \sin^{-1}\left(\frac{x}{5}\right) + \sin^{-1}\left(\frac{4}{5}\right) = \frac{\pi}{2}\) Now we see that the given equation itself is of the required identity. Hence, using the identity, we have \(\frac{x}{5} = \frac{4}{5}\)

Solve for x

To find the value of x, multiply both sides of the equation by 5: \(x = 4\) So, the answer is (b) 4.

Key concepts

Arc Functions

Arc functions, also known as inverse trigonometric functions, are crucial for solving equations where the angle is unknown, but the trigonometric ratio is given. The six main arc functions correspond to the six fundamental trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.

For example, the arc sine function, written as \(\sin^{-1}(x)\) or \(\arcsin(x)\), gives us the angle whose sine is \(x\). Similarly, \(\operatorname{cosec}^{-1}(x)\) or \(\arccosec(x)\) provides the angle whose cosecant is \(x\). In the context of our exercise, knowing that \(\sin^{-1}(a) + \operatorname{cosec}^{-1}(a) = \frac{\pi}{2}\), helps us solve for \(x\) when we have an equation involving these arc functions.

Sine and Cosecant Relationship

Understanding the relationship between sine and its reciprocal function, cosecant, is an essential aspect of trigonometry. The cosecant of an angle is defined as the reciprocal of the sine of that angle, which can be expressed as \(\csc(\theta) = \frac{1}{\sin(\theta)}\). In the exercise, this relationship allows us to transform the cosecant function into a sine function.

With the provided identity, \(\sin^{-1}(a) + \operatorname{cosec}^{-1}(a) = \frac{\pi}{2}\), we can directly find the angle corresponding to the given sine value by equating it to the complementary angle of the sine value derived from the cosecant function, as both will sum up to \(\frac{\pi}{2}\). This is what makes finding the value of \(x\) straightforward in this scenario.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values within their domains. These identities are invaluable tools for simplifying expressions and solving equations that involve trigonometry. Some of the most used trigonometric identities include the Pythagorean identities, sum and difference formulas, and reciprocal identities.

One of the identities used in the exercise is the complementary angle identity involving inverse sine and inverse cosecant functions: \(\sin^{-1}(a) + \operatorname{cosec}^{-1}(a) = \frac{\pi}{2}\). This relationship between the two functions provides a method to simplify and solve the equation by considering their complementary nature, as they add up to a right angle or \(\frac{\pi}{2}\) radians.

Solving Trigonometric Equations

Solving trigonometric equations is a fundamental skill in mathematics that often involves the application of various trigonometric identities and algebraic methods. The approach can range from substituting equivalent expressions to manipulating the equation to make use of symmetric properties or known identities.

In the given exercise, we applied the identity \(\sin^{-1}(a) + \operatorname{cosec}^{-1}(a) = \frac{\pi}{2}\) and leveraged the relationship between sine and its reciprocal function, cosecant. By recognizing that the given equation fits the identity, we isolate \(x\) and solve it directly without the need for more complex algebraic maneuvers. This technique simplifies what could be a complicated process, making it more accessible and easier to understand.