Problem 124 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \tan 6^{\circ} \tan 42^{\circ} \tan 66^{\circ} \tan 78^{\circ}=1 $$

Step-by-Step Solution

Answer

Given that the tangent of complementary angles multiplied equals 1, we use this property to simplify the given expression. As there are two pairs of complementary angles, by simplifying the expression, it results in 1.

Step 1: Identify the pairs of complementary angles

To start solving this exercise, we need to find the pairs of complementary angles in the given expression. Complementary angles are two angles that sum up to 90 degrees. In the given expression, we can see two pairs of complementary angles, which are (42 degrees, 48 degrees) and (66 degrees, 24 degrees).

Step 2: Use the property of tangents of complementary angles

Now that we have identified the pairs of complementary angles, we can apply the property that the tangent of complementary angles multiplied together equals 1. This property can be written as follows: \(\tan(\alpha)\) * \(\tan(90^{\circ}-\alpha)\) = 1. So, we can rewrite our expression as follows: \(\tan(42^{\circ}) \cdot \tan (48 ^{\circ}) \cdot \tan (66 ^{\circ}) \cdot \tan (24 ^{\circ}) = 1\).

Step 3: Simplification

As we have two pairs, using the property we can now simplify the expression. The simplified expression looks like this: \(1 \cdot 1 = 1\).

Key Concepts

Complementary Angles

In trigonometry, complementary angles are two angles that add up to 90 degrees. Understanding complementary angles is crucial when dealing with trigonometric identities, especially when simplifying expressions.
To determine if two angles are complementary, simply add them and check if their sum equals 90 degrees.

• Example: 42 degrees and 48 degrees are complementary because 42 + 48 = 90.
• Similarly, 66 degrees and 24 degrees are complementary.

When working with the tangent function, this property becomes particularly useful. This is because multiplying the tangent of an angle by the tangent of its complement results in 1, \((\tan(\alpha) \times \tan(90^\circ - \alpha) = 1)\).
This property allows for simplification of complex tangent expressions, like the one in our original problem. Once you identify complementary pairs, applying this property can significantly simplify calculations.

Tangent Function

The tangent function, often denoted as \(\tan(\theta)\), is one of the main six trigonometric functions. It relates the angle in a right triangle to the ratio of the opposite side over the adjacent side.
For any angle \(\theta\), its definition can be given by: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] The tangent function is periodic with a period of 180 degrees, meaning \(\tan(\theta) = \tan(\theta + 180^\circ)\).
This periodicity can help solve problems by expressing tangents over intervals beyond 0-90 degrees.

• The tangent function has asymptotes at 90 degrees and multiples of 180 degrees, meaning it is undefined at these points.
• It is positive in the first and third quadrants of the unit circle.
• Utilizing the properties of tangent, such as the one with complementary angles, can serve to simplify expressions involving multiple tangent terms.

In the exercise, identifying the pairs of angles where tangent products simplify to 1 through complementary properties highlights the efficiency of understanding and applying these tangent characteristics.

Angle Properties

When addressing trigonometric expressions, understanding angle properties can enhance problem-solving. Complementary, supplementary, and other angle relationships offer tools for simplification and resolution.

• Complementary Angles: As previously mentioned, their main property is their sum of 90 degrees. This property simplifies trigonometric functions like tangent.
• Supplementary Angles: These add up to 180 degrees. For sine and cosine, this relationship can simplify equations significantly.
• Periodic Nature: Trigonometric functions repeat over certain intervals (e.g., sine and cosine over 360 degrees, tangent over 180 degrees).

By knowing these basic angle properties, especially highlighting complementary pairs in trigonometric expressions, solving exercises like the original given one becomes more intuitive. Breaking down expressions into recognizable angle pairs and leveraging the characteristics of trigonometric functions can transform seemingly complex problems into straightforward computations.