Question

\(\log \tan 1^{\circ}+\log \tan 2^{\circ}+\ldots+\log \tan 89^{\circ}\) is : (a) 1 (b) \(1 / \sqrt{2}\) (c) 0 (d) \(-1\)

Short answer

a) 1 b) 45 c) 0 d) 89 Answer: c) 0

Step-by-step solution

Identify a key property of angles and tangent function

We notice that if we pair the angles from the start and end of the sum, they have complementary angles, meaning that the sum of the two angles is 90 degrees. The tangent function has the property \(\tan(90^\circ - x) = \cot x = \frac{1}{\tan x}\).

Write down pairs and apply tangent function property

Since we are given complementary angles, we can utilize the tangent property mentioned above. We can write the given expression in terms of pairs from the start and end of the sum: \(\log \tan 1^\circ + \log \tan 2^\circ +...+ \log \tan 89^\circ = \log \tan 1^\circ + \log \tan 89^\circ + \log \tan 2^\circ + \log \tan 88^\circ + ...\)

Use the logarithm property

Use the property of logarithms \(\log a + \log b = \log (a \times b)\) to combine the \(\log\) terms of complementary angle pairs: \((\log \tan 1^\circ + \log \tan 89^\circ) + (\log \tan 2^\circ + \log \tan 88^\circ) + ...= \log(\tan 1^\circ \times \tan 89^\circ) + \log(\tan 2^\circ \times \tan 88^\circ) + ...\)

Apply tangent property to simplify the pairs

Now, for each pair in the sum, apply the tangent property (\(\tan(90^\circ - x) = \frac{1}{\tan x}\)): \(\log(\tan 1^\circ \times \tan 89^\circ) + \log(\tan 2^\circ \times \tan 88^\circ) + ... = \log \left(\frac{1}{\tan 1^\circ} \times \tan 1^\circ \right) + \log \left (\frac{1}{\tan 2^\circ} \times \tan 2^\circ \right) + ...\)

Simplify the pairs and find the sum

We can simplify each term inside the logarithm, as they are in the form of \(\frac{1}{\tan x} \times \tan x\): \(\log \left(\frac{1}{\tan 1^\circ} \times \tan 1^\circ \right) + \log \left (\frac{1}{\tan 2^\circ} \times \tan 2^\circ \right) + ... = \log 1 + \log 1 + ...\) As \(\log 1 = 0\), the sum is just adding zeros: \( 0 + 0 + ... = 0\) So, the sum of the given expression \(\log \tan 1^\circ + \log \tan 2^\circ +...+ \log \tan 89^\circ\) is equal to 0. Therefore, the correct answer is (c) 0.

Key concepts

Logarithmic Properties

Logarithms have a special set of rules that make solving equations and simplifying expressions much easier. One key property is the log product rule:

• \( \log a + \log b = \log (a \times b) \)

This property allows us to combine the logarithms of two factors by multiplying them inside a single logarithm.
For instance, if you have \( \log 2 + \log 3 \), you can simplify this to \( \log (2 \times 3) \) or \( \log 6 \).
In our original exercise, since we are dealing with a series of tangent values combined with log functions, this property is essential. It allows for turning a long sum of logarithms into a product, which significantly simplifies calculations.

Complementary Angles

Complementary angles are two angles that add up to

• 90 degrees

They are often used in trigonometry for simplifying expressions due to their unique properties.
When dealing with trigonometric functions, knowing that two angles are complementary can help determine the value of trigonometric expressions. For example, the sine of an angle is the cosine of its complementary angle:

• \( \sin(90^\circ - x) = \cos(x) \)

This relationship also holds for the tangent function, where:

• \( \tan(90^\circ - x) = \cot(x) \)

In the context of our problem, using the complementary property of angles paired with the tangent and cotangent identities allows us to simplify each pair of terms, ultimately reducing their logarithmic expressions to zero.

Tangent Function

The tangent function, denoted as \( \tan(x) \), is one of the fundamental trigonometric functions and is vital in both geometry and calculus. The tangent function is defined as:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

Understanding how this function behaves for complementary angles is essential in trigonometry. For an angle \(x\) and its complementary angle \(90^\circ - x\), the tangent has a unique identity:

• \( \tan(90^\circ - x) = \cot(x) = \frac{1}{\tan(x)} \)

This identity is crucial when simplifying expressions involving \( \tan \) and \( \cot \).
In our particular exercise, this property helps turn the product \( \tan(x) \times \cot(x) \) into 1, simplifying the log terms to \( \log 1 \), which equals zero.

Angle Pairing

The concept of angle pairing involves selecting two angles whose trigonometric functions complement each other to simplify expressions. In many trigonometric problems, especially those involving sums over series, recognizing these pairs helps to simplify the calculation.
In the exercise, pairing angles at the start and the end of the series (like 1° and 89°, 2° and 88°) leverages complementary properties, simplifying expressions as each pair sums to 90°.
This strategy transforms a potentially complicated expression into something more manageable by exploiting trigonometric identities.

• Each pair simplifies to \( \frac{1}{\tan(x)} \times \tan(x) \)
• This equates to 1, simplifying the logarithmic expression of each pair to \( \log 1 \)

Leveraging these insights allows problems involving angle pairings to be approached with minimal complexity.