Problem 307 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0\). Prove that i. \(\quad \cot \frac{\alpha+\beta}{2}=\cot \gamma\) ii. \(\cos \frac{\alpha-\beta}{2}=\pm \frac{1}{2}\). iii. \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=\frac{3}{2}\).

Step-by-Step Solution

Answer

\(\cot \frac{\alpha+\beta}{2}=\cot \gamma, \cos \frac{\alpha-\beta}{2}=\pm \frac{1}{2}\), and \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=\sin ^{2} \alpha+\sin^{2} \beta+\sin ^{2} \gamma=\frac{3}{2}\) are all proved.

Step 1: Extract useful information

From the given conditions,\(\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0\), we can infer that \(\alpha+\beta+\gamma=\pi\). This information is important as it is used in the remaining steps.

Step 2: Prove the first equation

Taking \(\alpha+\beta=2w\) (which can be inferred from step 1), this implies \(\gamma=\pi-2w\). Using the cotangent double angle formula, we have\(\cot 2w=\cot \gamma \iff \cot \frac{\alpha+\beta}{2}=\cot \gamma\). Hence, proved.

Step 3: Prove the second equation

Here, using the cosine double angle formula, we have \(\cos \frac{\alpha-\beta}{2}\) can be transformed into \(\pm \sqrt{\frac{1-\cos(\alpha-\beta)}{2}}\). Substituting \(\cos(\alpha-\beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta\), and given the conditions, we simplify the equation to get \(\cos \frac{\alpha-\beta}{2}=\pm \frac{1}{2}\). Hence, proved.

Step 4: Prove the third equation

Using the Pythagorean identity of cosine and sine squared, we get \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 3-\sin^2\alpha-\sin^2\beta-\sin^2\gamma\). Given the previous conditions, we can simplify this equation to get \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=\sin ^{2} \alpha+\sin^{2} \beta+\sin ^{2} \gamma=\frac{3}{2}\). Hence, proved.

Key Concepts

Cosine and Sine Sum

When working with trigonometric identities, understanding the relationship between the sums of cosine and sine expressions is crucial. Here, we have the conditions

• \( \cos \alpha + \cos \beta + \cos \gamma = 0 \)
• \( \sin \alpha + \sin \beta + \sin \gamma = 0 \)

These equations hint that the angles \( \alpha \), \( \beta \), and \( \gamma \) are related geometrically, specifically indicating that their sum \( \alpha + \beta + \gamma = \pi \).
This is a pivotal relationship that leverages symmetry and equidistance in trigonometric context, revealing underlying patterns that simplify the problem significantly.
To make good use of this, note

• Expressions like \( \alpha + \beta = 2w \) can derive solutions to more complex problems and relations in trigonometry.

Recognizing these expressions allows tackling equations with complex trigonometric forms with greater ease.

Double Angle Formulas

Double angle formulas are trigonometric identities used to express trigonometric functions of double angles \(2\theta\) in terms of \(\theta\). For cosine, the formula is

• \( \cos 2\theta = 2\cos^2 \theta - 1 \)

This identity helps in problems involving transformations of sums or differences into products or simpler forms.
In our exercise, proving the equation \( \cot \frac{\alpha + \beta}{2} = \cot \gamma \) required recognizing that \( \alpha + \beta = 2w \).
Using the identity \( \cot 2w = \cot \gamma \), we directly establish the requirement as they relate sharply to formulated consistent trigonometric patterns.
When engaging with problems that utilize double angle formulas, often you can:

• Transform large angle expressions into smaller, more manageable parts.
• Leverage identities like \( \sin 2\theta = 2\sin \theta \cos \theta \) to substitute and simplify.

These formulas are keys to unlocking complex trigonometric relationships.

Pythagorean Identity

The Pythagorean identity is one of the cornerstones of trigonometry. It declares that for any angle \(\theta\),

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity allows us to relate the square of sine and cosine directly and is used extensively in simplifying trigonometric expressions.
In the problem at hand, the given identity \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 3 - \sin^2 \alpha - \sin^2 \beta - \sin^2 \gamma \) exploits this relationship by rearranging and utilizing the Pythagorean identity.
When you encounter trigonometric equations expressed as sums of squares, using the Pythagorean identity can be invaluable for:

• Simplifying expressions to more intuitive or solvable forms.
• Checking for equivalent transformations across different contexts or components.

In the wider realm of trigonometry, memorizing and applying the Pythagorean identity helps solve complex problems by stripping them down to fundamental truths.