Problem 25 - 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). \(\tan 305^{\circ}\)

Step-by-Step Solution

Answer

\(\tan 305^{\circ} = -\tan 55^{\circ}\).

Step 1: Identify the Quadrant

First, determine the quadrant in which the angle \(305^{\circ}\) lies. Angles in standard position start from the positive x-axis and rotate counterclockwise. Quadrant IV spans from \(270^{\circ}\) to \(360^{\circ}\). Since \(305^{\circ} > 270^{\circ}\) and \(305^{\circ} < 360^{\circ}\), it is in Quadrant IV.

Step 2: Determine the Reference Angle

The reference angle is the acute angle that the terminal side of the angle makes with the x-axis. In Quadrant IV, the reference angle is calculated by subtracting the angle from \(360^{\circ}\). Thus, the reference angle \(\theta_r = 360^{\circ} - 305^{\circ} = 55^{\circ}\).

Step 3: Determine the Tangent of the Reference Angle

Since the tangent function is positive in Quadrant I and Quadrant III, and negative in Quadrant IV, the sign of \(\tan 305^{\circ}\) should match. Calculate \(\tan 55^{\circ}\) since \(\tan 305^{\circ} = -\tan 55^{\circ}\) because it is in Quadrant IV.

Step 4: Express in terms of \(\tan 55^{\circ}\)

Conclude by expressing \(\tan 305^{\circ}\) in terms of its reference angle. \(\tan 305^{\circ} = -\tan 55^{\circ}\).

Key Concepts

Reference Angle

Understanding the concept of a reference angle is key to solving trigonometric problems. A reference angle is the smallest angle that a given angle makes with the x-axis. It is always a positive acute angle, meaning it measures less than 90 degrees. For any angle in standard position, the reference angle helps relate the trigonometric function of the original angle to that of a positive acute angle, simplifying calculations.

The process to determine a reference angle depends on the quadrant in which the angle lies:

• For angles in Quadrant I, the reference angle is the angle itself.
• In Quadrant II, calculate the reference angle by subtracting the given angle from 180°.
• In Quadrant III, subtract 180° from the given angle to obtain the reference angle.
• For Quadrant IV, subtract the angle from 360°.

By transforming complex angles into their reference angles, calculations become much simpler and more intuitive. With a reference angle, finding the trigonometric function values often just requires considering the sign based on the quadrant.

Tangent Function

The tangent of an angle in trigonometry is a fundamental concept. It is defined as the ratio of the opposite side to the adjacent side in a right triangle, or alternatively, as the ratio of the sine to the cosine of the angle: \[tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]

Depending on the angle's position, particularly in which quadrant it falls, the sign of the tangent function changes:

• Quadrant I: Tangent is positive.
• Quadrant II: Tangent is negative.
• Quadrant III: Tangent is positive again.
• Quadrant IV: Tangent is negative.

When working with larger angles or determining the tangent of an angle that's not within the typical range, using reference angles becomes very helpful. This allows you to calculate tangent based on the known values of these acute angles and adjust for the sign according to the quadrant.

Quadrants of Angles

The coordinate system in trigonometry is divided into four quadrants. Each quadrant dictates different signs for sine, cosine, and tangent functions. Knowing which quadrant an angle falls into helps in determining these sign changes.

Here's a quick guide to the quadrants:

• Quadrant I (0 to 90 degrees): All trigonometric functions are positive.
• Quadrant II (90 to 180 degrees): Sine is positive, while cosine and tangent are negative.
• Quadrant III (180 to 270 degrees): Tangent is positive, while sine and cosine are negative.
• Quadrant IV (270 to 360 degrees): Cosine is positive, while sine and tangent are negative.

When the angle \[305^{\circ}\] lands in Quadrant IV, as in the original exercise, it indicates that the tangent is negative due to the position and the signs associated with the quadrant. Understanding the quadrant positions provides clarity and simplifies the process of finding trigonometric values, including calculating reference angles and adjusting function signs appropriately.