Question

The value of \(x\) satisfying the equation \(\sin x+\frac{1}{\sin x}=\frac{7}{2 \sqrt{3}}\) is (a) \(10^{\circ}\) (b) \(30^{\circ}\) (c) \(45^{\circ}\) (d) \(60^{\circ}\)

Short answer

Question: Determine the value of x that satisfies the equation sin(x) + 1/sin(x) = 7/(2√3). Options: (a) 10° (b) 30° (c) 45° (d) 60° Answer: (d) 60°

Step-by-step solution

Evaluate the Sine Function for Each Angle

Calculate the sine function for each angle: \(\sin(10^{\circ})\approx 0.1736\) \(\sin(30^{\circ})= \frac{1}{2}\) \(\sin(45^{\circ})= \frac{1}{\sqrt{2}} \approx 0.7071\) \(\sin(60^{\circ})= \frac{\sqrt{3}}{2}\approx 0.8660\)

Calculate the Reciprocal of Each Sine Value

Calculate the reciprocal of each sine value: \(\frac{1}{\sin(10^{\circ})}\approx 5.758\) \(\frac{1}{\sin(30^{\circ})}= 2\) \(\frac{1}{\sin(45^{\circ})}= \sqrt{2} \approx 1.4142\) \(\frac{1}{\sin(60^{\circ})}= \frac{2}{\sqrt{3}} \approx 1.1547\)

Add the Sine and Reciprocal Sine Values Together

Add the sine value and reciprocal sine value for each angle: \(10^{\circ}: 0.1736 + 5.758 = 5.9316\) \(30^{\circ}: \frac{1}{2} + 2 = \frac{5}{2} = 2.5\) \(45^{\circ}: 0.7071 + 1.4142 = 2.1213\) \(60^{\circ}: \frac{\sqrt{3}}{2} + \frac{2}{\sqrt{3}} = \frac{7}{2\sqrt{3}}\)

Compare the Results and Select the Correct Answer

Compare the results from step 3 with the given value \(\frac{7}{2\sqrt{3}}\). The value that satisfies the given equation is the one for \(x=60^{\circ}\), as \(\frac{\sqrt{3}}{2} + \frac{2}{\sqrt{3}} = \frac{7}{2\sqrt{3}}\). The correct answer is: (d) \(60^{\circ}\)

Key concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, often abbreviated as 'sin'. It is defined for a right-angled triangle as the ratio of the length of the opposite side to the hypotenuse (the longest side). This ratio remains constant for a given angle, making the sine function angle-dependent.

The sine function is also defined for all real numbers using the unit circle, where the angle represents the arc length from the positive x-axis, and the sine of the angle is the y-coordinate of the corresponding point on the unit circle. Specifically, \(\sin(x)\) varies from -1 to 1 as x varies through the real numbers, with important values at \(0^\circ\), \(90^\circ\), \(180^\circ\), and \(270^\circ\) where the sine function respectively has values 0, 1, 0, and -1.

The sine function exhibits a periodic nature, meaning it repeats its values in regular intervals, specifically every \(360^\circ\) or \(2\pi\) radians.

Reciprocal of Sine

The reciprocal of sine, also known as cosecant (csc), is the multiplicative inverse of the sine function. For any angle x, the reciprocal is given by \(\csc(x) = \frac{1}{\sin(x)}\) as long as \(\sin(x) eq 0\).

The concept of the reciprocal is crucial when solving trigonometric equations. Just as the sine function has important properties, so does the cosecant. The reciprocal can dramatically change the behavior of the function, as seen with values that are near zero, where the reciprocal becomes very large. Furthermore, similar to sine, cosecant also has a periodic nature and undefined values where the sine function is zero, leading to vertical asymptotes on its graph.

Evaluating Trigonometric Functions

Evaluating trigonometric functions like the sine or its reciprocal involves calculating their values for specific angles. This can usually be done by using known values, unit circle definitions, or calculators for more complicated measurements. An understanding of important angle values such as \(30^\circ\), \(45^\circ\), and \(60^\circ\) and their respective sine values can be especially helpful. Important values such as \(\sin(30^\circ) = \frac{1}{2}\), \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\), and \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\) are often used in calculations.

In practice, evaluating a trigonometric function often involves computation or estimation techniques. For example, calculating the sine function first and then its reciprocal – as shown in the provided exercise – allows for stepwise simplification to approach the solution of trigonometric equations.

Trigonometric Identities

Trigonometric identities are equations that hold true for all eligible values of the involved angles. They are the backbone of manipulating and solving trigonometric equations. Some of the basic identities include the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\), the reciprocal identities such as \(\csc(x) = \frac{1}{\sin(x)}\), and the angle sum and difference identities.

Understanding trigonometric identities is critical when solving complex trigonometric equations. They provide tools for simplifying expressions, proving other mathematical properties, and converting between different trigonometric functions. In the context of the exercise, no complex identities were used, but simplification using these concepts is often necessary in more advanced problems.