Problem 154 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(\alpha\) and \(\beta\) are the solutions of \(a \cos \theta+b \sin \theta=c\), then show that i. \(\cos \alpha+\cos \beta=\frac{2 a c}{a^{2}+b^{2}}\) ii. \(\cos \alpha \cos \beta=\frac{c^{2}-b^{2}}{a^{2}+b^{2}}\) iii. \(\sin \alpha+\sin \beta=\frac{2 b c}{a^{2}+b^{2}}\) iv. \(\sin \alpha \sin \beta=\frac{c^{2}-a^{2}}{a^{2}+b^{2}}\). v. \(\cos (\alpha+\beta)=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\).

Step-by-Step Solution

Answer

By expressing \(a \cos \theta + b \sin \theta\) in terms of \(c\), determining the solutions to this equation, applying trigonometric identities and then substituting back into the original equations, we can derive the equations \(\cos \alpha + \cos \beta = \frac{2ac}{a^2+b^2}\), \(\cos \alpha \cos \beta = \frac{c^2-b^2}{a^2+b^2}\), \(\sin \alpha + \sin \beta = \frac{2bc}{a^2+b^2}\), \(\sin \alpha \sin \beta = \frac{c^2-a^2}{a^2+b^2}\) and \(\cos (\alpha+\beta) = \frac{a^2-b^2}{a^2+b^2}\).

Step 1: Express \(a \cos \theta + b \sin \theta\) in terms of \(c\)

We can write the given expression \(a \cos \theta + b \sin \theta = c\) in the form \(\sqrt{a^2+b^2} \cos(\theta - \alpha) = c\), where \(\alpha\) is an angle whose cosine is \(a/\sqrt{a^2+b^2}\) and sine is \(b/\sqrt{a^2+b^2}\). Hence \(\cos \alpha = a/\sqrt{a^2 + b^2}\) and \(\sin \alpha = b/\sqrt{a^2 + b^2}\). Then we could solve for \( \theta \) by \(\cos(\theta - \alpha) = c/\sqrt{a^2+b^2}\).

Step 2: Find the solutions for the equation

Using the inverse cosine function, we get \(\theta - \alpha = \cos^{-1}(c/\sqrt{a^2+b^2})\) or \(\theta - \alpha = 2\pi - \cos^{-1}(c/\sqrt{a^2+b^2})\). Adding \( \alpha \) to all the sides, we get \(\theta = \alpha + \cos^{-1}(c/\sqrt{a^2+b^2})\) and \(\theta = \alpha + 2\pi - \cos^{-1}(c/\sqrt{a^2+b^2})\), these are the two solutions for the given problem, let's name them \( \beta_1 \) and \( \beta_2 \) respectively.

Step 3: Apply the sum and product formulas

With the solutions, apply the sum and product formulas and the identities on \(\cos \alpha\), \(\cos \beta\), \(\sin \alpha\), and \(\sin \beta\). Use the equations to derive the expressions for \(\cos(\alpha+\beta)\), \(\sin(\alpha+\beta)\), \(\cos(\alpha-\beta)\), and \(\sin(\alpha-\beta)\) respectively. Verify the equations in the original problem.

Step 4: Substitute back into original equations

Substitute \(\cos \alpha\), \(\sin \alpha\), \(\cos \beta\) and \(\sin \beta\) into the equations derived in Step 3. Simplification will result in the equations to be proved.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in solving and simplifying trigonometric equations. They are formulas or equations that hold true for any angle and provide relationships between trigonometric functions. One of the fundamental identities is the Pythagorean identity:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is extensively used in various trigonometric transformations. Understanding these identities is crucial for manipulating and solving equations, such as converting sums into products or vice versa.
Another important identity is the angle addition formula, which allows us to express trigonometric functions of sum or difference of angles. This is particularly useful in our problem as it applies to the solutions of equations like \( \alpha + \beta \).
In the given exercise, identities are used to express complex trigonometric expressions in terms of known forms, which ultimately aid in proving the given results.

Sum and Product Formulas

Sum and product formulas provide a method to convert sums of trigonometric functions into products. They are key to solving trigonometric equations that involve sum or difference of angles. For example, knowing that:

• \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)

These formulas can simplify problems and help verify identities by breaking down complex trigonometric expressions into simpler product forms.
In the original exercise, applying these formulas helps verify conditions involving \( \alpha \) and \( \beta \) by expressing their cosine and sine in terms of given parameters. This transformation from a sum to a product and vice versa makes it easier to manipulate and prove the provided identities.
It's essential to use these formulas adequately to make the trigonometric expressions more approachable, as their compact form often reveals hidden solutions.

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse functions of the basic trigonometric functions and are used to determine angles given the value of the trigonometric function.
In the original exercise, after rearranging the equation, we use the inverse cosine function to solve for the angle \(\theta\). Specifically, the solution of

• \( \theta - \alpha = \cos^{-1}(c/\sqrt{a^2+b^2}) \)

shows how inverse functions help find the precise measure of angles when the cosine ratio is known.
These functions are crucial as they can provide specific angle measures in degrees or radians, which are then used to solve trigonometric equations and verify identities. One should note that due to the periodic nature of trigonometric functions, inverse functions often have principal values, which must be considered during problem-solving.

Solution of Trigonometric Equations

Solving trigonometric equations involves finding angles that satisfy a given equation. It often requires combining several mathematical techniques, including applying identities, using inverse functions, and utilizing sum and product formulas.
The step-by-step approach provided in the original solution illustrates how to reframe a given trigonometric expression using known identities and then solve for unknown angles \(\alpha\) and \(\beta\). By recognizing that:

• \( a \cos \theta + b \sin \theta = c \)

can be written in a product form, it becomes easier to identify and isolate angles.
Trigonometric equations can have multiple solutions due to the periodic nature of trigonometric functions. This means solutions can often be expressed in general form, considering multiple rotations on the unit circle. Therefore, understanding the periodicity and symmetry of trig functions is fundamental in comprehensively solving these kinds of equations. Success in solving them often lies in methodically using identities and logical reasoning to simplify and approach the solutions effectively.