Problem 69 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If in a \(\triangle A B C, a \sin A=b \sin B\), then show that the triangle is isosceles.

Step-by-Step Solution

Answer

Given \(a \sin A = b \sin B\), we rewrite it using sine rule: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Since \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), there are two possibilities: 1) angle A = angle B, 2) angle A = angle C or angle B = angle C. Analyzing both possibilities, it is proven that the triangle ABC is isosceles.

Step 1: Rewrite the given equation using sine rule

In any triangle, the sine rule states that \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\) Given: \(a \sin A = b \sin B\), we will rewrite the equation using sine rule: \(\frac{a^2}{a \sin A} = \frac{b^2}{b \sin B}\)

Step 2: Simplify and substitute using sine rule

Now we can simplify the equation: \[\frac{a}{\sin A} = \frac{b}{\sin B}\] Since \(\frac{a}{\sin A} = \frac{b}{\sin B}\), using the sine rule, we get: \[\frac{c}{\sin C} = \frac{a}{\sin A} = \frac{b}{\sin B}\]

Step 3: Analyze the possibilities

There are two possibilities that would make the triangle isosceles: 1. Angle A = Angle B 2. Angle A = Angle C or Angle B = Angle C

Step 4: Prove that angle A = angle B or angle A = angle C

Using the given equation \(a \sin A = b \sin B\) and sine rule, let's analyze the possibilities. 1. If angle A = angle B, then \(\sin A = \sin B\) and using the sine rule: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). This possibility holds true since the conditions are already met. 2. If angle A = angle C or angle B = angle C, then consider when A = C. We will have \(\sin A = \sin C\). From sine rule, we have \(\frac{a}{\sin A} = \frac{c}{\sin C}\). Substituting \(\sin A\) with \(\sin C\), we get \(\frac{a}{\sin A} = \frac{c}{\sin A}\), which yields \(a=c\). The triangle becomes isosceles with A = C. Similarly, if B = C, we would have \(\sin B = \sin C\) and then \(\frac{b}{\sin B} = \frac{c}{\sin C}\). Substituting \(\sin B\) with \(\sin C\), we get \(\frac{b}{\sin B} = \frac{c}{\sin B}\), which yields \(b=c\). The triangle becomes isosceles with B = C.

Step 5: Conclusion

Therefore, based on the given condition \(a \sin A = b \sin B\), it is proven that the triangle ABC is isosceles, either angle A = angle B, angle A = angle C or angle B = angle C.

Key Concepts

Sine Rule

In the realm of trigonometry, the Sine Rule is a powerful tool used for solving triangles. This rule connects the lengths of the sides of a triangle to the sines of its angles. Specifically, it asserts that in any triangle

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

where \(a, b,\) and \(c\) are the lengths of the sides opposite to angles \(A, B,\) and \(C\) respectively.
This relationship helps establish a constant of proportionality among these ratios. When you know any three of these quantities, the Sine Rule can guide you to find the fourth.
It's particularly handy in situations involving non-right triangles, including isosceles triangles, where at least two sides and angles are equal. This makes it a versatile equation in solving and proving properties of such triangles.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between angles and sides of triangles. It centers around a few critical functions, namely sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions relate the angles in a triangle to the ratio of its sides, providing a link between geometry and algebra.

• **Sine** (\(\sin\)) is defined as the ratio of the length of the opposite side of an angle to the hypotenuse.
• **Cosine** (\(\cos\)) is the ratio of the adjacent side to the hypotenuse.
• **Tangent** (\(\tan\)) is the ratio of the opposite side to the adjacent side.

Beyond simple definitions, trigonometry allows applying complex mathematical concepts to solve real-world problems, including proving the properties of isosceles triangles. Whether for calculating distances or understanding wave behavior, trigonometry is an essential mathematical tool.

Triangle Properties

Triangles come with a rich assortment of properties, often interconnected, making them a fascinating study in geometry. One key property is related to their angles. The sum of the angles in any triangle always equals 180 degrees.
This foundational fact applies whether the triangle is scalene, isosceles, or equilateral. In the case of **isosceles triangles**, there are specific properties to note:

• At least two sides are of equal length.
• Angles opposite these equal sides are also equal.

These symmetries make isosceles triangles easier to manage and predict, particularly when utilizing the Sine Rule or other trigonometric principles. Understanding these properties not only aids in solving geometric problems but also in providing proofs, such as demonstrating whether a triangle is isosceles based on given side lengths and angle relationships.