Problem 20 - Algebra 2 and Trigonometry

Question

In \(18-27\) , express each given function value in terms of a function value of a positive acute angle (the reference angle). \(\tan 255^{\circ}\)

Step-by-Step Solution

Answer

\(\tan 255^{\circ} = \tan 75^{\circ}\)

Step 1: Identify the Quadrant

First, determine in which quadrant the angle \(255^{\circ}\) lies. Angles between \(180^{\circ}\) and \(270^{\circ}\) belong to the third quadrant.

Step 2: Determine the Reference Angle

The reference angle \(\theta_{ref}\) is the positive acute angle formed with the x-axis. For angles in the third quadrant, use the formula \(\theta_{ref} = \theta - 180^{\circ}\). For \(\theta = 255^{\circ}\), calculate \(\theta_{ref} = 255^{\circ} - 180^{\circ} = 75^{\circ}\).

Step 3: Apply the Tangent Sign Rule

In the third quadrant, tangent is positive. Therefore, the value of \(\tan 255^{\circ}\) is \(\tan 75^{\circ}\).

Step 4: Final Expression

Express \(\tan 255^{\circ}\) in terms of the reference angle: \(\tan 255^{\circ} = \tan 75^{\circ}\).

Key Concepts

Understanding the Tangent Function

The tangent function, often denoted as \( \tan(\theta) \), is one of the basic trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right triangle. In terms of the unit circle, it is the ratio of the sine to the cosine, that is, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This function is periodic with a period of \(180^{\circ}\) or \(\pi\) radians, which means the value of the function repeats after every \(180^{\circ}\). This periodicity is crucial in simplifying angles, such as \(255^{\circ}\), into their equivalent positive acute angles using reference angles.

One key aspect of the tangent function is its sign, which depends on the quadrant where the angle lies. In the first and third quadrants of the unit circle, tangent values are positive, while in the second and fourth quadrants, they are negative. Recognizing this pattern is important for correctly simplifying tangent expressions.

The Third Quadrant and Its Influence

Angles in the third quadrant range from \(180^{\circ}\) to \(270^{\circ}\). In this quadrant, both sine and cosine values turn negative, but because tangent is defined as the ratio of sine to cosine, these negative values cancel each other out. As a result, tangent remains positive in the third quadrant. This is why the third quadrant is significant when simplifying trigonometric expressions.

For instance, an angle like \(255^{\circ}\) lies in the third quadrant. By calculating its reference angle, we can simplify the computation. Here, the reference angle forms by subtracting \(180^{\circ}\) from \(255^{\circ}\), giving \(75^{\circ}\). Knowing \(75^{\circ}\) is positive and acute helps us express the original tangent as \(\tan(75^{\circ})\). This understanding aids in dealing with complex trigonometric problems.

Utilizing Trigonometric Identities Effectively

Trigonometric identities are formulas that relate the angles and ratios of trigonometric functions. They are immensely useful for simplifying trigonometric functions like the tangent. One commonly used identity is the Pythagorean identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\). This identity relates sine and cosine and can be manipulated to derive other forms of tangent expressions.

Another important identity involves the tangent function itself and helps when working across different quadrants. It utilizes the periodicity of the tangent function: \(\tan(\theta + 180^{\circ}) = \tan(\theta)\). This identity directly informs us how to transform an angle like \(255^{\circ}\) to its reference angle. Additionally, remembering the symmetry of the tangent across the origin, given \(\tan(-\theta) = -\tan(\theta)\), assists in calculating tangent for negative angles.

• It's crucial to practice these identities to simplify complex trigonometric expressions efficiently.
• Understanding these deeper relationships between functions enhances problem-solving skills in trigonometry.