Problem 116 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

In the ambiguous case, given \(a, b, A\), prove that the sum of the areas of the two triangles formed is \(\frac{1}{2} b^{2} \sin 2\)

Step-by-Step Solution

Answer

The sum of the areas of the two triangles that can be formed in the ambiguous case using sides \(a\) and \(b\) and angle \(A\) is \(\frac{1}{2} b^{2} \sin 2A\). This is proved by calculating the areas individually using the formula for the area of a triangle \(\frac{1}{2} ab \sin A\), then adding the two areas together and simplifying using the double-angle formula for sine.

Step 1: Identify the triangles

The triangle given by the ambiguous case can result in either one triangle, two different triangles or no triangle based on the angle size. But here we will consider two triangles are formed.

Step 2: Calculate the area of the first triangle

We calculate the area of first triangle using \(\frac{1}{2} ab \sin A\)

Step 3: Calculate the area of the second triangle

The area of the second triangle will be \(\frac{1}{2} ab \sin (180-A)\), since the base and height are the same but the angle is supplementary to the original. Hence, \(\sin (180-A) = \sin A\) as sine function of an angle and its supplementary angle are equal.

Step 4: Sum the areas and simplify

We need to add the areas of two triangles. This is \(\frac{1}{2} ab\sin A + \frac{1}{2} ab\sin A = ab\sin A = \frac{1}{2} ab \(2 \sin A \cos A\) = \frac{1}{2} b^{2} \sin 2A\), using the double-angle formula for sine.

Key Concepts

Triangle Area Calculation

The area calculation of triangles in trigonometry is a fundamental concept that aids in understanding shapes and their properties.
An important formula to calculate the area of a triangle when involving angles, especially in the context of trigonometry is:

• Area = \( \frac{1}{2} \times base \times height \)
• Area = \( \frac{1}{2} ab \sin A \) when two sides and the included angle are known

Understanding how to calculate the area when given an angle allows solving problems involving shapes where side measurements aren't straightforward.
The application of the area formula by angles, as seen in the exercise, is pivotal when dealing with ambiguous cases, where multiple triangles may exist with the given angles and sides.

Sine Rule

The Sine Rule is a crucial tool in solving problems related to triangles, particularly when determining unknown sides or angles.
It is given by the formula:

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

This rule applies to any type of triangle, whether it's acute, obtuse, or right-angled.
By using the Sine Rule, students can efficiently find missing measurements and resolve ambiguity in certain cases—like the ambiguous case, where the same angle could result in two different triangle configurations.
It's important to note that the Sine Rule assists in identifying whether a given set of triangle information corresponds to one, two, or no possible triangle.

Double Angle Formula

The Double Angle Formula is a significant feature in trigonometry that allows the calculation of trigonometric functions of double angles based on single angles.
For sine, it's expressed as:

• \( \sin 2A = 2 \sin A \cos A \)

The clarity in using the Double Angle Formula can simplify complex trigonometric expressions and make solving related equations more manageable.
In the given solution, the formula was used to simplify the sum of triangle areas, converting from basic sine function to a double angle.

• This helps in reducing the complexity and enables clearer understanding through simplified expressions.

This technique is useful when angles are doubled and you need to find simplified expressions for solving geometry problems.

Trigonometry in Triangles

Understanding trigonometry within the context of triangles unlocks a deeper comprehension of geometry and mathematical relationships.
Trigonometry specifically deals with the relationship between the angles and sides of triangles, offering a profound toolset for problem-solving.

• It uses identities, rules, and formulas like the Sine Rule and Double Angle Formula to resolve complex triangle problems.
• This branch of mathematics frequently resolves ambiguous cases where various conditions provide different solutions yet yield workable results.

In applications like the exercise above, trigonometry helps deduce the existence and characteristics of triangles from partial or unclear data.
Students benefit from mastering trigonometry in triangles, gaining skills to logically approach and solve advanced mathematical challenges involving triangular shapes.