Problem 12 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

In a tringle \(\Delta A B C\), prove that \(\frac{\sin B}{\sin C}=\frac{c-a \cos B}{b-a \cos C}\)

Step-by-Step Solution

Answer

By applying the Law of Sines and the Law of Cosines, it is proven that \(\frac{\sin B}{\sin C}=\frac{c-a \cos B}{b-a \cos C}\) for the triangle \( \Delta ABC \).

Step 1: Identify Given and Required

We are given a triangle \( \Delta ABC \) and an equation, \(\frac{\sin B}{\sin C}=\frac{c-a \cos B}{b-a \cos C}\). The task is to prove this equation in terms of the Law of Sines and the Law of Cosines.

Step 2: Use the Law of Cosines

The Law of Cosines states that for any triangle with sides of length a, b and c and the angle between the sides a and b is C (let), then: \(c^2=a^2+b^2-2ab \cos C\). Substitute the values of \(a^2\), \(b^2\), and \(2ab \cos C\) with \(c^2\), rearranging gives us \(c = a\cos B + b\cos C\)

Step 3: Use the Law of Sines to Substitute

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides and angles. Thus, we can write \(\frac{a}{\sin A} = \frac{b}{\sin B}=\frac{c}{\sin C}\). Rearrange these equations for a, b, and c, respectively, then substitute these into the equation we got in Step 2.

Step 4: Simplify the equation

Substitute the values found in Step 3 into the equation from Step 2, then simplify the equation. The equation will transform into the required equation, which proves that \(\frac{\sin B}{\sin C} =\frac{c-a \cos B}{b-a \cos C}\).

Key Concepts

Law of Sines

The Law of Sines is a vital tool in trigonometry, particularly useful when working with non-right triangles. It helps us relate the sides and angles of any triangle. According to the Law of Sines, the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles in the triangle. This can be mathematically expressed as:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

Here, \(a\), \(b\), and \(c\) are the lengths of the sides, and \(A\), \(B\), and \(C\) are the opposite angles. This law is particularly useful when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). It allows you to find unknown lengths or angles in the triangle. When solving triangle problems, especially those aiming to prove identities like the one in the original exercise, the Law of Sines helps establish relationships between the angles and sides easily.

Law of Cosines

The Law of Cosines extends the Pythagorean theorem to apply to all triangles, not just right triangles. It's particularly handy when dealing with triangles when you have two sides and the included angle (SAS) or all three sides (SSS). The formula is:

• For a triangle with sides \(a\), \(b\), and \(c\) and angle \(C\): \( c^2 = a^2 + b^2 - 2ab \cos C \)

This is helpful in situations where you need to find a side length when you know the other two sides and the included angle or vice versa. In our specific exercise, the Law of Cosines is used to express one of the triangle's sides in terms of the known sides and angles. By rearranging the formula, we can derive expressions for the cosine of an angle using side lengths, facilitating further simplification and leading us to the proof of the given equation.

Trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where the functions are defined. The fundamental identities include:

• Pythagorean Identities, like \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Angle Sum and Difference Identities, for example, \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
• Double Angle Identities, such as \( \sin 2A = 2 \sin A \cos A \)

These identities are the building blocks for solving trigonometric equations and proving complex relationships in triangle problems. In problems like the one posed, where you need to manipulate trigonometric expressions, having a strong grasp of these identities allows for simplification and transformation of the given equations into their required forms. Whether you're proving equations or simplifying expressions, these identities are essential tools in your trigonometry toolkit.