Problem 22 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a \sin x+b \cos (x+\theta)+b \cos (x-\theta)=d\), then the minimum value of \(|\cos \theta|\) is (A) \(\frac{1}{2|a|} \sqrt{d^{2}-a^{2}}\) (B) \(\frac{1}{2|b|} \sqrt{d^{2}-a^{2}}\) (C) \(\frac{1}{2|b|} \sqrt{a^{2}-d^{2}}\) (D) none of these

Step-by-Step Solution

Answer

The minimum value of \(|\cos \theta|\) is (B) \(\frac{1}{2|b|} \sqrt{d^2-a^2}\).

Step 1: Simplify the Expression

Starting from the given equation: \[ a \sin x + b \cos(x+\theta) + b \cos(x-\theta) = d \]Use the angle sum and difference identities: \[ \cos(x+\theta) = \cos x \cos \theta - \sin x \sin \theta \]\[ \cos(x-\theta) = \cos x \cos \theta + \sin x \sin \theta \]Adding these two, we get: \[ 2b\cos x\cos \theta \]Therefore, the equation becomes: \[ a\sin x + 2b \cos x \cos \theta = d \]

Step 2: Express in Standard Form

We want to write the equation in terms of a single trigonometric function:\[ a \sin x + 2b \cos x \cos \theta = d \]Factor out the common factor for easier analysis:\[ \sqrt{a^2 + (2b \cos \theta)^2} \left( \frac{a}{\sqrt{a^2+(2b\cos \theta)^2}} \sin x + \frac{2b \cos \theta}{\sqrt{a^2+(2b\cos \theta)^2}} \cos x \right) = d \]This is of the form \( R \sin(x + \alpha) \) where \( R = \sqrt{a^2+(2b\cos \theta)^2} \).

Step 3: Simplify with Trigonometric Identity

Using the identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) and equate:\[ \left( \frac{a}{\sqrt{a^2+(2b\cos \theta)^2}} \right)^2 + \left( \frac{2b \cos \theta}{\sqrt{a^2+(2b\cos \theta)^2}} \right)^2 = 1 \]

Step 4: Analyze Minimum Condition

Since the minimum value of \( \sin(x + \alpha) \) can be \( -1 \), the minimum value of the expression is:\[ -\sqrt{a^2+(2b\cos \theta)^2} = d \]Squaring both sides:\[ a^2 + 4b^2 \cos^2 \theta = d^2 \]Solve for \( \cos^2 \theta \):\[ \cos^2 \theta = \frac{d^2 - a^2}{4b^2} \]

Step 5: Compute Minimum \(|\cos \theta|\)

Take the square root on both sides to find minimum \(|\cos \theta|\):\[ |\cos \theta| = \frac{1}{2|b|} \sqrt{d^2 - a^2} \]Thus, the minimum value is option (B):\[ \boxed{\frac{1}{2|b|} \sqrt{d^2-a^2}} \]

Key Concepts

Minimum Value

When solving trigonometric problems, especially those concerning expressions like \(a \sin x + b \cos(x + \theta) + b \cos(x - \theta) = d\), determining the minimum value of certain components is crucial. In this context, we aim to find the minimum value of \(|\cos \theta|\). This involves analyzing the expression after simplifying it using trigonometric identities.

The minimum value is derived from the expression \(2b \cos x \cos \theta\). If you aim to find when an expression, say \(\sin(x + \alpha)\), reaches its minimum, you should consider its basic range. The sine function has a minimum of \(-1\).

• This insight is used by setting \(-R = d\) while transforming the equation into the form \(R\sin(x + \alpha) = d\).
• Here, \(R\) is a derived constant given by \(\sqrt{a^2+(2b\cos \theta)^2}\).

By squaring and solving for \(\cos^2 \theta\), we find that \(|\cos \theta| = \frac{1}{2|b|} \sqrt{d^2 - a^2}\). Understanding this, helps us appreciate how trigonometric identities aid in simplifying problems and extracting meaningful results.

Angle Sum and Difference Identities

Angle sum and difference identities are powerful tools for simplifying trigonometric expressions. In the exercise \(a \sin x + b \cos(x + \theta) + b \cos(x - \theta) = d\), we utilize these identities to transform the expression.

These identities take the form:

• \( \cos(x+\theta) = \cos x \cos \theta - \sin x \sin \theta \)
• \( \cos(x-\theta) = \cos x \cos \theta + \sin x \sin \theta \)

Adding these results simplifies the expression to \(2b \cos x \cos \theta\), making it easier to handle. Such simplification allows us to factor the equation efficiently while conserving its integrity.

Understanding how these identities work encourages breaking down complex expressions into manageable parts. Once broken down, expressions become easier to analyze and solve, highlighting the elegance of trigonometry in transforming problems.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions often involves a combination of identities and algebraic manipulation. As seen in our exercise, once we've used the angle identities, we further simplify the expression to bring it into a form manageable for solving.

After applying the angle sum and difference identities, the expression in the form \(a\sin x + 2b\cos x \cos \theta = d\) is still quite complex. By introducing common factors and transforming it into a standard form, we make the expression easier to interpret:

• Factor out terms and form a \(R\sin(x + \alpha)\) expression.
• Utilize Pythagorean identities to equate and verify expressions like \((a/\sqrt{a^2+(2b\cos \theta)^2})^2 + (2b\cos \theta/\sqrt{a^2+(2b\cos \theta)^2})^2 = 1\).

Finally, through this methodical simplification, we can derive useful values like those of \(|\cos \theta|\). Mastering these manipulations is vital for tackling a wide array of trigonometric problems efficiently, showcasing the art of trigonometry in expression simplification.