Problem 48 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If in a triangle \(A B C\), the median \(A D\) and the perpendicular \(A E\) from the vertex \(A\) to the side \(B C\) divides the angle \(A\) into three equal parts, show that \(\cos \left(\frac{A}{3}\right) \cdot \sin ^{2}\left(\frac{A}{3}\right)=\frac{3 a^{2}}{32 b c}\).

Step-by-Step Solution

Answer

On analyzing and solving, we found that if the median and perpendicual from a vertex of a triangle divide the angle into three equal parts, then \(\cos \left(\frac{A}{3}\right) \cdot \sin^{2}\left(\frac{A}{3}\right)=\frac{3 a^{2}}{32 b c}\).

Step 1: Label the Triangle

Label the triangle as follows: \(A B C\) is the main triangle with \(D\) being the point on \(B C\) where the median from \(A\) meets \(B C\), and \(E\) as the foot of the perpendicular from \(A\) to \(B C\). Let \(B A = a\), \(A C = b\) and \(C B = c\). Let \(A D = m\), \(A E = p\), \(B E = q\) and \(E C = r\). The triangle is thus divided into 3 parts with equal angles \(\frac{A}{3}\).

Step 2: Formulate the equations

Construct the equations using trigonometric identities and the given data: From the right triangle \(AEB\), \(q = p \tan \frac{A}{3}\). And from the triangle \(AED\), \(m = p \sec \frac{A}{3}\). Using the formula for median \(m = \sqrt{\frac{2b^{2}+2c^{2}-a^{2}}{4}}\), substitute \(m\) and \(p\) in terms of \(a\), \(b\), and \(c\).

Step 3: Solve and simplify

After substituting, we get \[p = a \sqrt{\frac{b^{2}+c^{2}-a^{2}}{8bc}}\] and \[q = a \tan \frac{A}{3} \sqrt{\frac{b^{2}+c^{2}-a^{2}}{8bc}}\]. Solving these two equations will give the value for \(\cos \left(\frac{A}{3}\right) \cdot \sin^{2}\left(\frac{A}{3}\right)\).

Step 4: Final Calculation

We can then use the double angle identity for cosine, \[cos^{2} x - sin^{2} x = cos2x = 1 - 2 sin^{2} x \implies (cos x) \cdot (sin^{2} x) = \frac{1}{2}(1 - cos2x)\]. By replacing \(x\) with \(\frac{A}{3}\) and substituting the values of p and q, we can simplify to find that \(\cos \left(\frac{A}{3}\right) \cdot \sin^{2}\left(\frac{A}{3}\right)=\frac{3 a^{2}}{32 b c}\).

Key Concepts

Median of a Triangle

In the realm of triangles, a median holds special importance. A median in a triangle connects a vertex to the midpoint of the opposite side. It essentially splits the triangle into two smaller triangles of equal area. This helps in various calculations and understanding the properties of a triangle. For example, if you have triangle \(ABC\) and draw a median from vertex \(A\) to the side \(BC\), the point where the median meets \(BC\) is called \(D\). Here, \(AD\) is the median.

• The median doesn't necessarily divide the angle at the vertex into equal parts.
• It does, however, always split the opposite side into segments of equal length.
• This property becomes useful in many geometric proofs and problems.

Thus, understanding medians can enhance your problem-solving intuition significantly by providing symmetry and balance within a triangle.

Trigonometric Identities

Trigonometric identities are like the building blocks when it comes to solving complex trigonometry problems. They help you transform and simplify expressions involving angles and sides of triangles. Common trigonometric identities include:

• Sum and difference identities
• Double angle identities
• Half angle identities

The double angle identity for cosine, for instance, tells us that \( \cos2x = \cos^2x - \sin^2x \). Another way to express it is \( 1 - 2\sin^2x \) which is very handy in calculations. These identities provide tools that simplify trigonometric functions to make problems more manageable, especially when dealing with non-standard angles. Understand and memorize these identities, and you'll find many problems become simpler to solve. They are like shortcuts or bridges that help us navigate through the twists and turns of trigonometric problems.

Angle Trisection

Trisection of an angle means dividing an angle into three equal parts. Classic geometry problems often use angle trisection to create precise divisions in constructions. While trisecting angles using only a compass and straightedge is impossible in general, there are specific instances where trisection is feasible.

• Some angles are trivially easy to trisect, like 90 degrees which gives three 30 degree angles.
• In the given triangle problem, angle \(A\) is divided into three equal parts by constructing a median and a perpendicular.
• These conditions allow us to use specific trigonometric properties to find relationships between the sides and angles.

Understanding angle trisection provides valuable insight when handling complex problems. Remember, it is the specific geometric constraints that enable us to perform such operations, which are invaluable even in real-world applications, like dividing a plot of land precisely or designing a structure with exact angles.