Problem 41 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

In a triangle \(A B C\), if \(\cos A \cdot \cos B+\sin A \sin B \sin C\) \(=1\), prove that the sides are in the ratio \(1: 1: \sqrt{2}\).

Step-by-Step Solution

Answer

The sides of the triangle ABC have the ratio `1:1:√2`.

Step 1: Understand the given condition

The condition given in the problem is: \(\cos A \cdot \cos B+\sin A \sin B \sin C =1\). This needs to be used to derive the relation between the sides of the triangle.

Step 2: Apply Trigonometric Identities

Applying the identity \(\cos A \cos B = 1 - \sin^2 A - \sin^2 B\), the given equation can be rewritten as \(\sin^2 A + \sin^2 B - \sin A \sin B \sin C = 0\). Use this equation for further simplification.

Step 3: Apply Law of Sines

According to the Law of Sines, we know that \(\sin A/a = \sin B/b = \sin C/c\), where a, b, c are the sides opposite to respective angles A, B and C. Use this fact to substitute \(\sin A = a/a\) and \(\sin B = b/a\) in the equation derived in Step 2.

Step 4: Simplify the equation

After substitution in Step 3, the equation becomes \(a^2 + b^2 - abc\sin C/a^2 = 0\). The equation can be rearranged to give \(a^2(b+c - a) = 0\), which gives us \(a = 0, a = b+c\). Since, in a triangle side length cannot be zero, we discard first solution. Thus, we get \(a = b+c\)

Step 5: Evaluate the ratio required

Using the condition \(a=b+c\), we can clearly see that the ratio between the sides of the triangle ABC is '1: 1: √2'. This can be obtained by considering the sides of the triangle to be 1, 1 and √2 respectively. This set of side lengths satisfy the condition \(a = b+c\) and hence the given condition.

Key Concepts

Triangles

Triangles are fundamental shapes in geometry, characterized by having three sides, three angles, and three vertices. In adding up the interior angles, you will always find the sum equals 180 degrees. Triangles can be classified based on their angles or sides:

• Equilateral Triangle: All sides and angles are equal.
• Isosceles Triangle: Two equal sides and angles.
• Scalene Triangle: No equal sides or angles.

Understanding different types of triangles helps tackle many problems in trigonometry, like those involving specific side ratios or angle relationships.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions like sine, cosine, and tangent. They are true for every value of the variable and are a powerful tool for simplifying expressions. Some of the key identities include:

• Pythagorean Identities: For example, \( an^2\theta+1=\sec^2\theta\).
• Angle Sum and Difference: Such as \( an(a \pm b)=\frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}\).

In the context of the problem, identities like \( an A \) or \( an B \) are used to rearrange and simplify the given conditions to derive new relationships.

Law of Sines

The Law of Sines is a powerful trigonometric rule that relates the sides and angles of a triangle. It's given by the formula: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]where \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively. This law is particularly useful when dealing with non-right triangles:

• It helps find unknown angles when side lengths are known.
• You can also find unknown sides when two angles and one side are known.

In the exercise, it's used to express relationships between the sides and angles, helping to prove specific conditions about the triangle's ratios.

Triangle Side Ratios

Triangle side ratios are important in determining specific types of triangles and solving trigonometric problems. A well-known set of ratios is \(1:1:\sqrt{2}\). These ratios occur in specific types of right-angled triangles called 45-45-90 triangles:

• This means the two legs of the triangle are equal, and the hypotenuse is \sqrt{2}\ times the length of a leg.
• Such a ratio simplifies problems involving diagonal measurements.

In the given problem, recognizing this ratio helps to confirm the conditions for the triangle, using the derived relations from steps like \(a = b + c\), allowing us to see the triangle sides fit this special ratio.