Question

Let \(a, b, c\) be three non-zero real numbers such that the equation $$ \sqrt{3} a \cos x+2 b \sin x=c, x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $$ has two distinct real roots \(\alpha\) and \(\beta\) with \(\alpha+\beta=\frac{\pi}{3}\). Then, the value of \(\frac{b}{a}\) is

Short answer

\(\frac{b}{a} = \frac{1}{\sqrt{3}}\)

Step-by-step solution

- Express the equation in terms of r and theta

Use the identity \(a\cos(x) + b\sin(x) = r\cos(x - \theta)\), where \(r = \sqrt{a^2 + b^2}\) and \(\theta = \arctan(\frac{b}{a})\). Here, we first find the values of \(r\) and \(\theta\) for the given values of \(a\) and \(b\), which are \(\sqrt{3}a\) and \(2b\) respectively.

- Write the trigonometric expression in cosine form

We rewrite the given equation \(\sqrt{3} a \cos x + 2 b \sin x = c\) in the form of \(r\cos(x - \theta)\), which is \(r\cos(x - \theta) = c\).

- Apply the cosine of sum identity for roots

The equation \(r\cos(x - \theta) = c\) must hold for the two distinct roots \(\alpha\) and \(\beta\). Since these roots satisfy the equation, we can write \(r\cos(\alpha - \theta) = c\) and \(r\cos(\beta - \theta) = c\).

- Use the sum of roots

Given \(\alpha + \beta = \frac{\pi}{3}\), we find the cosine of the average of the two roots. This can be represented as \(\cos\left(\frac{\alpha + \beta}{2} - \theta\right)\), which simplifies to \(\cos\left(\frac{\pi}{6} - \theta\right)\). Since \(\alpha\) and \(\beta\) are distinct roots and their sum is given, the equation would have a maximum at \(x = \frac{\alpha + \beta}{2}\).

- Determine the value of theta

The maximum value of \(\cos\) is 1, which occurs when its argument is zero. Therefore, \(\frac{\pi}{6} - \theta = 0\), which implies \(\theta = \frac{\pi}{6}\).

- Calculate b/a

With \(\theta = \arctan(\frac{b}{a})\), and \(\theta = \frac{\pi}{6}\), we can set up the equation \(\frac{b}{a} = \tan(\frac{\pi}{6})\). Solving this, we find that \(\frac{b}{a} = \tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}}\).

Key concepts

Trigonometric Identities

Trigonometric identities are equalities involving trigonometric functions that are true for every value of the involved variables where both sides of the equality are defined. These identities are indispensable when simplifying trigonometric expressions or solving trigonometric equations. A fundamental example is the Pythagorean identity, \(\sin^2(x) + \cos^2(x) = 1\), which relates the squares of the sine and cosine of an angle.

When faced with a linear combination of sine and cosine like in the exercise, \(a\cos(x) + b\sin(x)\), we often use another identity that allows us to combine these terms into a single trigonometric function: \(a\cos(x) + b\sin(x) = r\cos(x - \theta)\), with \(r\) representing the magnitude and \(\theta\) representing the phase shift. Here, \(r\) is calculated as \(\sqrt{a^2 + b^2}\), and \(\theta\) is \(\arctan(\frac{b}{a})\). This method simplifies the process of finding the roots of the equation.

Sum of Angles in Trigonometry

The sum of angles formula is an important concept that helps in simplifying the evaluation of trigonometric functions of the sum or difference of two angles. Specifically, the cosine of the sum of two angles is given by the formula \(\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\).

In our problem, knowing that the sum of the two roots \(\alpha + \beta\) is \(\frac{\pi}{3}\), we can use this sum formula indirectly to find the cosine of the average of these two angles, which is necessary to determine the roots’ relationship with the equation’s coefficients. This formula exemplifies the importance of the sum of angles in solving trigonometric equations.

Cosine of Sum Formula

The cosine of sum formula, \(\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\), is particularly useful when dealing with trigonometric equations that involve the cosine of a sum or difference of two angles.

In our exercise, by applying this formula, we are able to express the given equation in terms of a single cosine function and thus, make the relationship between the equation's coefficients and the roots of the equation more evident. This ultimately leads to discerning the value of \(\frac{b}{a}\) as it relates to the angle \(\theta\) found in the problem.

Real Roots of Trigonometric Equations

The real roots of trigonometric equations are the solutions that satisfy the equation within its domain, which, in a real-world context, correspond to the angles for which the trigonometric equation holds true. Finding real roots often involves understanding the graphical representation of trigonometric functions and applying identities to simplify the equations first.

In the context of the given equation, \(\sqrt{3} a \cos x+2 b \sin x=c\), the real roots can be found by expressing the equation in a form that consolidates the trigonometric functions into a single cosine function. The ability to evaluate the cosine function at various points is essential in determining the unique solutions that constitute the real roots. In this case, the exercise leads us to utilize the sum of roots and the cosine of sum formula to eventually find the ratio of \(\frac{b}{a}\).

Understanding the concept of real roots is crucial because it bridges our understanding of trigonometric equations from abstract theory to practical applications, and aides in solving real-world problems involving wave patterns, oscillations, and other phenomena describable by trigonometric functions.