Question

The value of \(\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}\) is (a) 2 (b) 3 (c) 4 (d) 0

Short answer

#Question# Evaluate the expression \(\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}\). #Answer# The given expression can be simplified to \(-\tan 54^{\circ}\), but this value does not match any of the provided options. Therefore, there is no correct answer from the given options.

Step-by-step solution

Recognize trigonometric identities

In this problem, we need to use the following trigonometric identities: \(tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\) \(tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\) Essentially, we will break down the given expression into pairs and apply these identities.

Group the terms in pairs

First, group the terms in pairs: \(\tan 9^{\circ}-\tan 27^{\circ} = \tan(9^{\circ}-27^{\circ})\) \(\tan 63^{\circ}+\tan 81^{\circ} = \tan(63^{\circ}+81^{\circ})\) Regarding the first pair, since it's a difference, we'll use the first identity (difference of tangents). For the second pair, as it's a sum, we'll use the second identity (sum of tangents).

Apply the identities

To find the value of each pair, apply the corresponding identity: For the difference pair: \(\tan(9^{\circ}-27^{\circ}) = \frac{\tan 9^{\circ} - \tan 27^{\circ}}{1 + \tan 9^{\circ} \tan 27^{\circ}}\) Note that \(9^{\circ}-27^{\circ}=-18^{\circ}\). For the sum pair: \(\tan(63^{\circ}+81^{\circ}) = \frac{\tan 63^{\circ} + \tan 81^{\circ}}{1 - \tan 63^{\circ} \tan 81^{\circ}}\) Note that \(63^{\circ}+81^{\circ}=144^{\circ}\).

Simplify the expressions

Now, let's simplify the expressions: \(\tan(-18^{\circ}) = -\tan 18^{\circ}\) (since tangent is odd function, \(\tan(-x)=-\tan x\)) \(\tan(144^{\circ}) = \tan(144^{\circ} - 180^{\circ}) = -\tan 36^{\circ}\) (since tangent has a period of \(180^{\circ}\), \(\tan(x+180^{\circ})=\tan x\)) The expression now becomes: \(-\tan 18^{\circ} - \tan 36^{\circ}\) The problem is now a difference of tangents. Applying the difference identity again: \(-\tan 18^{\circ} - \tan 36^{\circ} = \tan(-18^{\circ}-36^{\circ}) = \tan(-54^{\circ})\)

Simplify the final expression

Using the property \(\tan(-x) = -\tan x\), we find the final result: \(\tan(-54^{\circ}) = -\tan 54^{\circ}\) Now, we compare this result with the given options: (a) 2 (b) 3 (c) 4 (d) 0 The value of \(-\tan 54^{\circ}\) is not any of the above integers. Hence, no option is the correct answer.

Key concepts

Tangent Function

The tangent function is one of the six fundamental trigonometric functions, which also include sine, cosine, secant, cosecant, and cotangent. It is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. In terms of sine (\text{sin}) and cosine (\text{cos}), it is expressed as:

\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
This function is periodic, with a period of \(180^\circ\) or \(\pi\) radians, meaning that \(\tan(\theta + 180^\circ) = \tan(\theta)\). In addition, it is an odd function, which implies that \(\tan(-\theta) = -\tan(\theta)\). These properties play a crucial role when simplifying trigonometric expressions, such as the one in the exercise. Also, the tangent function can have values that range from negative to positive infinity, and it is undefined for angles where cosine is zero (e.g., \(90^\circ\), \(270^\circ\)).

Understanding how the tangent function behaves and how it interacts with other trigonometric functions is invaluable when attempting to simplify complex expressions or solve trigonometric equations.

Angle Sum and Difference Identities

Angle sum and difference identities are pivotal tools in trigonometry. They allow for the simplification of expressions involving the sine and cosine of the sum or difference of two angles. The tangent function specifically has the following sum and difference identities:

\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
These formulas demonstrate how to decompose the tangent of an angle sum or difference into an expression involving the tangents of individual angles. During the exercise, these identities were used to break down the given sum into components that could be individually solved. The process of grouping and re-grouping the angles is crucial to simplify the problem into more manageable parts—highlighting the importance of these identities in computational trigonometry.

Trigonometric Functions Properties

Trigonometric functions exhibit various properties that make them predictable and useful in solving numerous geometrical and algebraic problems. A fundamental property is their periodic nature; sine and cosine functions have periods of \(360^\circ\) or \(2\pi\) radians, while as mentioned earlier, tangent and cotangent have periods of \(180^\circ\) or \(\pi\) radians. Trigonometric functions also demonstrate symmetry. The sine and tangent functions are odd, where \(\sin(-\theta) = -\sin(\theta)\) and \(\tan(-\theta) = -\tan(\theta)\), while the cosine function is even, which shows that \(\cos(-\theta) = \cos(\theta)\). Furthermore, the periodic and symmetric nature of these functions can lead to simplifications such as \(\tan(\theta + 180^\circ) = \tan(\theta)\). These fundamental properties were applied in the solution to re-express and calculate the given tangent expressions. Recognizing and applying these properties can significantly ease the process of solving trigonometric problems.