Problem 101 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \tan ^{2} 5^{\circ}=\tan 25^{\circ} $$

Step-by-Step Solution

Answer

Assuming the initial statement \( \tan^{2} 5^{\circ} = \tan 25^{\circ} \) is false, one suggested correction could be \( \tan 5^{\circ} = \tan 5^{\circ} \) or \( \tan 25^{\circ} = \tan 25^{\circ} \). Another suggested true statement could be \( \tan^{2} 5^{\circ} = \sin^2 25^{\circ} \) or \( \tan^{2} 5^{\circ} = \cos^2 25^{\circ} \).

Step 1: Evaluate \(\tan^{2} 5^{\circ}\)

Calculating the tangent of 5 degrees and then squaring it gives the value of the left side of the equation.

Step 2: Evaluate \(\tan 25^{\circ}\)

Calculating the tangent of 25 degrees gives the value of the right side of the equation.

Step 3: Compare Both Sides

Compare the value from step 1 and step 2 to check if both values are equal. If they are not equal, then the statement is false.

Step 4: Correcting the Statement

Assuming the initial statement is false, the correct statement would be \( \tan 5^{\circ} = \tan 5^{\circ} \) or \( \tan 25^{\circ} = \tan 25^{\circ} \). Another possible true statement could be \( \tan^{2} 5^{\circ} = \sin^2 25^{\circ} \) or \( \tan^{2} 5^{\circ} = \cos^2 25^{\circ} \). The exact corrected statement should be calculated based on trigonometric identities and known values.

Key Concepts

Tangent Function

The tangent function, usually denoted as \(\tan(\theta)\), is one of the basic trigonometric functions. It expresses the ratio between the opposite side and the adjacent side of a right triangle with angle \(\theta\). In more general terms, in the unit circle, it corresponds to the y-coordinate divided by the x-coordinate at a given angle from the positive x-axis.

Mathematically, it can be expressed as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] or
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]

This shows how closely related the tangent function is to sine and cosine, which are also fundamental trigonometric functions. The tangent function is periodic, with a period of 180 degrees (or \(\pi\) radians), meaning that \(\tan(\theta + 180^\circ) = \tan(\theta)\). This is helpful in identifying patterns and symmetries within trigonometric equations.

Since the tangent function relies on division by the cosine, it is undefined wherever the cosine is zero, which happens at odd multiples of 90 degrees (e.g., 90, 270 degrees, etc.). Thus, we should be cautious to avoid these values when working with the tangent to prevent division by zero errors.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all possible values of the involved angles. They are useful for simplifying expressions and solving trigonometric equations.

Some common trigonometric identities include:

• **Pythagorean Identities**:
  \[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
• **Quotient Identities**:
  \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
• **Reciprocal Identities**:
  \[ \sec(\theta) = \frac{1}{\cos(\theta)}, \quad \csc(\theta) = \frac{1}{\sin(\theta)}, \quad \cot(\theta) = \frac{1}{\tan(\theta)} \]

These identities allow us to rewrite certain trigonometric expressions in different forms that might be more suitable for specific problems.

In the original exercise, it’s beneficial to use these identities to correct the false statement. For example, using the relation \( \tan^2(\theta) + 1 = \sec^2(\theta) \), one can generate alternative equations that are true for any \(\theta\). This helps in balancing equations where combining or decomposing functions lead to equivalent valid expressions. Understanding these identities thus provides a strong basis for handling trigonometric problems effectively.

Degrees in Trigonometry

Degrees are a unit of measurement for angles in trigonometry. Typically, a full circle is divided into 360 equal parts, each part being one degree. This measurement is practical for many trigonometric calculations and applications, especially in geometry and navigation.

In trigonometry, degrees are often used interchangeably with radians, another unit for measuring angles where \(360^\circ = 2\pi\) radians. While radians are mathematically simpler in calculus and higher-level mathematics, degrees are intuitive and visually more understandable for everyday problem-solving.

For example, the conversion between degrees and radians is given by:

• \(180^\circ = \pi\) radians
• \(1^\circ = \frac{\pi}{180}\) radians
• Converting \(\theta\) degrees to radians: \(\theta^\circ \times \frac{\pi}{180}\)

Using degrees facilitates easier understanding and direct computation of angles, as demonstrated in the original exercise where \(\tan(5^\circ)\) and \(\tan(25^\circ)\) were evaluated. Observing trigonometric angles in degrees also helps students grasp the periodic nature and frequency of trigonometric functions much faster, particularly when engaged in introductory studies.