Problem 49 - Algebra 2

Question

Which pair of values are NOT equal? $$ \begin{array}{ll}{\text { A. } \tan \frac{\pi}{4},-\tan \frac{3 \pi}{4}} & {\text { B. } \tan \frac{\pi}{4}, \tan \frac{5 \pi}{4}} \\ {\text { C.tan } \theta,-\tan (-\theta)} & {\text { D. } \tan \theta, \tan (\pi-\theta)}\end{array} $$

Step-by-Step Solution

Answer

Option D is the pair that is not equal.

Step 1: Evaluate Option A

The tangent of \(\frac{\pi}{4}\) returns 1. The tangent of \(-\frac{3\pi}{4}\) also equals 1, due to the periodicity and symmetry of tangent function. So, these two are equal.

Step 2: Evaluate Option B

The tangent of \(\frac{\pi}{4}\) is 1. The tangent of \(\frac{5\pi}{4}\) also equals 1, because of the periodicity of tangent function. So, these two are equal.

Step 3: Evaluate Option C

For tangent function, \(\tan(\theta) = -\tan(-\theta)\). So, this pair is equal.

Step 4: Evaluate Option D

The tangent of \(\theta\) does not necessarily equal the tangent of \(\pi-\theta\). These are only equal if theta equals \(\frac{\pi}{4}\). So, in general these two are not equal.

Key Concepts

Periodicity of Trigonometric Functions

Trigonometric functions, like the tangent, repeat their values in a predictable way. This repetition is called periodicity. For the tangent function, the period is

• \(\pi\), meaning it repeats its pattern every \(\pi\) radians.

This property helps us evaluate expressions like \(\tan(\frac{\pi}{4})\) and \(\tan(\frac{5\pi}{4})\). Both give the same result because \((\frac{5\pi}{4} - \frac{\pi}{4}) = \pi\), which is one complete period. This is why knowing the period of trigonometric functions can simplify many calculations. It helps us see the relationship and equality between angles that differ by multiples of \(\pi\). Periodicity allows us to deduce that angles such as these lead to the same tangent value.

Tangent Angle Identities

The tangent function has specific mathematical identities that describe its properties and relationships with angles. Essential identities include:

• \(\tan(\theta) = -\tan(-\theta)\)
• \(\tan(\theta + \pi) = \tan(\theta)\)

These identities explain the results seen in options like C, where \(\tan(\theta)\) is equal to \(-\tan(-\theta)\). This is due to the odd nature of the tangent function, where changing the sign of the angle results in changing the sign of the tangent as well, but the magnitude remains the same. Another powerful identity is that \(\tan(\theta)\) repeats every \(\pi\) radians, reinforcing periodicity and explaining outcomes like those mentioned in options A and B. Understanding these identities aids in predicting tangent values without direct calculation, making problem solving faster.

Theta and Angle Relationships

Angles, when combined in certain ways, can lead to interesting identities and relationships among trigonometric functions. Theta (\(\theta\)) often appears in expressions like \(\pi - \theta\) and \(\pi + \theta\):

• \(\tan(\pi-\theta) = -\tan(\theta)\)
• These expressions reflect symmetry in angles on the unit circle.

In option D, \(\tan(\theta)\) doesn't match \(\tan(\pi-\theta)\), unless \(\theta\) itself is \(\frac{\pi}{4}\). This discrepancy highlights how closely the structure of angles affects tangent values. The difference arises because \(\pi - \theta\) represents reflection across the x-axis, altering the sign, but not necessarily aligning to equal original tangent values. Remembering these relationships aids in understanding when tangent values will change and when they will remain consistent.