Download Important RD Sharma Class 11 Solutions for Chapter 11 - Trigonometric Equations Exercise 11.1 - Free PDF
FAQs on RD Sharma Class 11 Solutions Chapter 11 - Trigonometric Equations
1. How do you find the principal solutions of a trigonometric equation according to the methods in RD Sharma Class 11 Chapter 11?
To find the principal solutions for a trigonometric equation, you need to find all solutions within the interval [0, 2π]. The method involves:
- First, solve the equation to find the value of the trigonometric function (e.g., sin x = 1/2).
- Identify the quadrants where the function has the given sign (e.g., sine is positive in the first and second quadrants).
- Determine the reference angle (acute angle) for the value.
- Calculate the angles in the identified quadrants. For sin x = 1/2, the principal solutions are π/6 and 5π/6.
RD Sharma solutions provide detailed examples for each function to master this process.
2. What is the step-by-step method to find the general solution for an equation like tan θ = tan α?
The general solution includes all possible angles that satisfy the equation. For tan θ = tan α, the steps are based on the periodicity of the tangent function:
- The basic condition is that θ and α must have the same tangent value.
- This occurs every π radians (or 180°).
- Therefore, we can write the relationship as θ = nπ + α, where 'n' is any integer (n ∈ Z).
- This single formula covers all possible solutions, as explained with proofs in RD Sharma Chapter 11.
3. How do the RD Sharma solutions explain solving equations of the form a cos x + b sin x = c?
This is a standard form, and RD Sharma solutions outline a clear method to solve it:
- Divide the entire equation by √(a² + b²).
- The equation becomes (a/√(a² + b²)) cos x + (b/√(a² + b²)) sin x = c/√(a² + b²).
- Let a/√(a² + b²) = cos α and b/√(a² + b²) = sin α. The equation simplifies to cos(x - α) = c/√(a² + b²).
- Now, solve this simpler trigonometric equation for (x - α) and then find the general solution for x.
Note that a solution exists only if |c/√(a² + b²)| ≤ 1.
4. Why is it incorrect to simply cancel a common trigonometric factor (like sin x) from both sides of an equation?
It is a common mistake to cancel a term like sin x because doing so eliminates potential solutions. When you divide by a variable term, you implicitly assume it is not zero. However, the values of x for which that term is zero (e.g., sin x = 0) could be valid solutions to the original equation. The correct method is to move all terms to one side and factor out the common term. For example, instead of cancelling sin x from 'sin x cos x = sin x', you should write 'sin x cos x - sin x = 0' and then factorise to get sin x (cos x - 1) = 0. This gives two sets of solutions: sin x = 0 and cos x = 1.
5. How does understanding the periodicity of trigonometric functions help in finding the general solution?
Periodicity is the fundamental concept behind general solutions. Since trigonometric functions repeat their values at regular intervals (the period), a single equation like sin x = 1/2 has infinite solutions. The period of sin x and cos x is 2π, while for tan x it is π. The general solution formula (e.g., x = nπ + (-1)ⁿα for sin x = sin α) uses an integer 'n' to represent these infinite repetitions. Essentially, 'n' allows you to 'jump' from one solution to the next by adding multiples of the period, ensuring no solution is missed.
6. When solving trigonometric equations, why is it crucial to check for extraneous solutions?
Extraneous solutions are values that emerge from the solving process but do not satisfy the original equation. They typically appear under two conditions:
- Squaring both sides: This can introduce new solutions because it eliminates negative signs. For instance, solving x = 2 gives x² = 4, but solving x² = 4 gives x = 2 and x = -2.
- Using identities with domain restrictions: If you use an identity involving tan x or sec x, you must ensure the solutions do not make the denominator zero (e.g., cos x ≠ 0).
Always substitute your final answers back into the original equation to verify they are valid.
7. How can trigonometric identities be strategically used to simplify and solve complex equations in Chapter 11?
Trigonometric identities are essential tools for simplifying complex equations into a solvable form. The strategy involves:
- Reducing to a single function: Use identities like sin²x + cos²x = 1 to convert the entire equation into a quadratic in either sin x or cos x.
- Reducing to a single angle: Use double-angle (e.g., sin 2x = 2 sin x cos x) or half-angle formulas to make all arguments in the equation the same (e.g., all 'x' instead of a mix of 'x' and '2x').
- Using sum-to-product formulas: Identities like sin C + sin D can convert sums into products, which is useful for setting the equation to zero and solving each factor separately.
The key is to recognise the structure of the equation and choose the identity that simplifies it most effectively.
