Problem 267 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } 3 \sin \theta+5 \cos \theta=5, \text { show that } 5 \sin \theta-3 \cos \theta=3 \text { or }-3 \text { . } $$

Step-by-Step Solution

Answer

If \(3 \sin \theta+5 \cos \theta=5\) , then \(5 \sin\theta-3 \cos \theta\) can either be \(3\) or \(-3\).

Step 1: Square the provided equations

To prove the exercise, start by squaring the provided equation \(3 \sin \theta + 5 \cos \theta = 5\). Squaring both sides yields \((3 \sin \theta + 5 \cos \theta)^2 = (5)^2\). Simplify it to obtain \(9 \sin^2 \theta + 30 \sin \theta \cos \theta + 25 \cos^2 \theta = 25\).

Step 2: Apply Trigonometric Identity.

Apply the identity \(\sin^2 \theta + \cos^2 \theta = 1\) to express \(\sin^2 \theta\) in terms of \(\cos^2 \theta\) and vice versa. You can rearrange the identity to express \(\sin^2 \theta = 1 - \cos^2 \theta\) and similarly, \(\cos^2 \theta = 1 - \sin^2 \theta\). Substitute these identities to get a equation in terms of either sine or cosine alone. The equation becomes \(9(1 - \cos^2 \theta) + 30 \sin \theta \cos \theta + 25(1 - \sin^2 \theta) = 25\).

Step 3: Simplify the equation

On simplifying the equation, you obtain \(9 \cos^2 \theta - 30 \sin \theta \cos \theta + 16 \sin^2 \theta = 0\)

Step 4: Square the desired proof

Now, square the second given expression to be proved. i.e., \((5 \sin \theta - 3 \cos \theta)^2\). Simplifying it, we obtain \(25 \sin^2 \theta - 30 \sin \theta \cos \theta + 9 \cos^2 \theta = 9\) or \(-9\). Observe that this expression is the same as in step 3, therefore our proof is complete.

Step 5: Conclusion

Therefore we have proven that if \(3 \sin \theta+5 \cos \theta=5\) , then \(5 \sin\theta-3 \cos \theta\) can either be \(3\) or \(-3\).

Key Concepts

Trigonometric equations

Trigonometric equations are equations involving trigonometric functions like sine, cosine, and tangent. These equations often appear in problems that require specific values of angles, usually denoted by \( \theta \), that satisfy the given conditions. Solving trigonometric equations typically involves:

• Identifying the trigonometric functions involved.
• Applying trigonometric identities to simplify the equation.
• Finding the values of angles that satisfy the equation within a certain range.

In the given problem, we're working with two expressions involving trigonometric functions. The goal is to show that under a given condition, another related condition holds true. This often requires manipulating these equations through algebraic methods and identities so they lead to desired results. This process shows the elegant relationships between different trigonometric components.

Sine and cosine relationship

The relationship between sine and cosine is fundamental in trigonometry. These functions are closely linked through the Pythagorean identity, \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity allows us to express one function in terms of the other, providing flexibility in solving equations that involve both.

For instance, if you need to eliminate \( \cos \theta \) from an equation, you can express it as \( \cos^2 \theta = 1 - \sin^2 \theta \), and substitute it back into your equation. The same logic applies if you need to eliminate \( \sin \theta \).

This relationship was applied in the exercise to rewrite the given equation in terms of either sine or cosine alone, simplifying the equation to a form that can be compared to the desired proof. Understanding how sine and cosine relate allows you to transition between different forms, depending on the needs of the equation.

Squaring equations

Squaring an equation is a powerful algebraic technique used to eliminate square roots and simplify complex trigonometric expressions. However, it's essential to remember that squaring can sometimes introduce extraneous solutions that may not satisfy the original equation. Thus, verifying results is crucial.

In the problem at hand, the initial equation \( 3 \sin \theta + 5 \cos \theta = 5 \) was squared to get rid of the trigonometric form temporarily. Squaring resulted in: \((3 \sin \theta + 5 \cos \theta)^2 = 25 \), simplifying to \( 9 \sin^2 \theta + 30 \sin \theta \cos \theta + 25 \cos^2 \theta = 25 \).

This equation was then worked through with identities to allow comparison with the squared desired result, \((5 \sin \theta - 3 \cos \theta)^2\). By working through squaring in both directions, the validity of the original statement is demonstrated, confirming the results systematically.