Problem 15 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(A\) and \(B\) are acute \(+\) ve angles satisfying the equations \(3 \sin ^{2} A+2 \sin ^{2} B=1\) and \(3 \sin 2 A=2 \sin 2 B\), then find \((A+2 B)\).

Step-by-Step Solution

Answer

To find the numerical value of \(A + 2B\), first solve the equations to find \(A\) and \(B\), and then compute \(A + 2B\). It is complex and involves the utilization of multiple trigonometric identities and principles. The step-by-step solution offers a detailed breakdown of the process.

Step 1: Identify the given equations

The two provided equations are: \(3 sin^{2} A + 2 sin^{2} B =1 \) and \(3 sin 2A = 2 sin 2B \). The goal is to find the sum \(A + 2B\).

Step 2: Equate the expressions

In the second equation, to eliminate either \(A\) or \(B\), make the sines equal to each other. This gives: \(sin 2A = sin 2B \). In this step, ensure you remember the sine function's positivity rule: in the first and second quadrants (0 to 180 degrees), the sine function remains positive.

Step 3: Apply General Solution of Sin Function

From the general solution of sin, we know that if sinY=sinZ then Y=n$\pi$ ± Z, where n is any integer. From Step 2, you know sin2A=sin2B. Now, applying the general solution gives us that 2A=n$\pi$ ± 2B.

Step 4: Elicit the possible solutions

Step 3 results in two subcases to consider: 1) 2A=n$\pi$ + 2B and 2) 2A=n$\pi$ - 2B. But since A and B are acute, both lie within 0 to 90 degrees, so they can not be negative. Hence, only the positive case holds true, resulting in: A=n$\pi$/2 + B.

Step 5: Substitute the value of A in Equation 1

Substitute A from Step 4 into Equation 1 to get to a single variable equation: \(3 sin^{2}(n\pi/2 + B) + 2 sin^{2} B = 1\).

Step 6: Simplify the equation

Using the identity \(sin (a + b) = sin a cos b + cos a sin b\) and Squaring the Term, simplify the term to find the value of \(B\).

Step 7: Compute the value of B

Solve the simplified equation in Step 6 to get the value of \(B\).

Step 8: Compute the value of A

Use the equation from Step 4 to compute the value of \(A\) using the already computed value of \(B\).

Step 9: Compute the value of (A+2B)

Finally, use the values of \(A\) and \(B\) found above to compute the value of \(A + 2B\).

Key Concepts

General Solution of Sine

In trigonometry, the general solution of the sine function is extremely helpful when solving equations involving periodic functions. Let's say we have the trigonometric equation \( \sin x = \sin y \) where \( x \) and \( y \) represent angles. This equation is true for multiple values of \( x \) because the sine function is periodic, repeating its values every \( 2\pi \) radians (or 360 degrees).

The general solution can be expressed as: \[ x = n\pi + (-1)^n y \] where \( n \) is an integer representing the number of times the sine wave has completed a full period (\pi radians) plus any additional angle that gives the same sine value. This concept is crucial when dealing with trigonometric equations, especially when the solutions are not restricted to a specific interval but extend to infinite angles.

• For angles in the first quadrant (acute angles), the general solution ensures the sine values are positive.
• In contexts with specified ranges (like acute angles), the general solution guides which values of \( n \) to consider.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved angles. These identities are the backbone of solving complex trigonometric equations and simplifying expressions.

One of the most commonly used identities is the Pythagorean identity, which states that \( \sin^2\theta + \cos^2\theta = 1 \), where \( \theta \) is an angle. Moreover, identities related to angle sums, like \( \sin(a + b) = \sin a \cos b + \cos a \sin b \) and \( \cos(a + b) = \cos a \cos b - \sin a \sin b \) are pivotal when dealing with angles expressed as the sum of two parts.

• Using identities helps to simplify equations into a form where the values of unknown angles can be calculated.
• Identities can also be used to derive other important equations relevant to specific problems.

Acute Angles in Trigonometry

Acute angles are those that measure less than 90 degrees (or \( \pi/2 \) radians) and are found in the first quadrant of the unit circle in trigonometry. For acute angles, all the trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—are positive.

When dealing with equations involving acute angles, it's crucial to consider the inherent properties of these angles:

• The sine and cosine of an acute angle are always between 0 and 1.
• Understanding these restrictions helps in predicting the range of possible solutions for trigonometric equations.

Acute angles frequently occur in real-life contexts, like the angles of elevation and depression, making them an important focus in trigonometry.

Solving Trigonometric Equations for Acute Angles

The process of solving trigonometric equations for acute angles involves several steps, which require using both trigonometric identities and properties of acute angles.

To solve for an acute angle:\

• It's essential to manipulate equations so that they contain only one trigonometric function. This can involve using identities or algebraic manipulation.
• After simplifying the equation, the range of possible angles is limited to \( (0, \pi/2) \) due to the nature of acute angles.
• Remember that for acute angles, the specific values of trigonometric functions like \(( \sqrt{1/2} \), \(( \sqrt{2/2} \), or \(( \sqrt{3/2} \)) correspond to well-known angles that can easily be identified.

By combining general solutions and angle identities with the unique properties of acute angles, one can solve trigonometric equations within a specific interval and find the desired angles.