Problem 24 - Algebra 2

Question

Find the exact value of each trigonometric function. \(\tan 300^{\circ}\)

Step-by-Step Solution

Answer

\(-\sqrt{3}\)

Step 1: Understand the Angle

First, we need to understand where the angle 300° is located on the unit circle. The angle 300° is in the fourth quadrant because it is between 270° and 360°.

Step 2: Reference Angle Calculation

The reference angle for 300° is calculated by subtracting 300° from 360°, which gives us 60°. Therefore, the reference angle for 300° is 60°.

Step 3: Identify the Sign of Tangent

In the fourth quadrant, the tangent function is negative. So, an(300°) will be negative.

Step 4: Use the Reference Angle

The tangent of the reference angle (60°) is known to be \( an(60^{\circ}) = \sqrt{3}\). Therefore, \(\tan(300^{\circ})\) is simply the negative of \(\tan(60^{\circ})\).

Step 5: Calculate the Exact Value

Since \(\tan(60^{\circ}) = \sqrt{3}\), and the sign of tangent in the fourth quadrant is negative, \(\tan(300^{\circ}) = -\sqrt{3}\).

Key Concepts

Unit Circle

The concept of the Unit Circle is an essential foundation in understanding trigonometric functions. Imagine a circle with a radius of one, centered at the origin (0,0) of a coordinate plane. This simple circle can help define the sine, cosine, and tangent functions for all possible angle measures.

The unit circle is crucial because:

• Each point on the circle corresponds to an angle, starting at 0° on the positive x-axis and moving counter-clockwise.
• It helps visualize angles greater than 360° or negative angles by continuing the rotation around the circle.
• On the unit circle, the horizontal coordinate (x) represents the cosine of the angle, and the vertical coordinate (y) represents the sine of the angle.

Learning the unit circle enriches your understanding of the relationships between angles and their trigonometric values.

Reference Angle

A reference angle is a useful tool when working with angles on the unit circle. It is the closest distance between the given angle and the x-axis. Reference angles are always positive and can be measured in degrees or radians.

Here’s why reference angles matter:

• They simplify calculations by reducing complex angles to simple, known trigonometric values.
• Finding a reference angle involves determining how far an angle is from 0°, 90°, 180°, 270°, or 360°, depending on the quadrant.
• For example, the reference angle for 300° is calculated as 360° - 300° = 60°, which is crucial in solving trigonometric problems.

With reference angles, you understand and find trigonometric function values more effortlessly.

Tangent Function

The Tangent Function is one of the primary trigonometric functions, aside from sine and cosine. It is defined as the ratio of the sine and cosine of an angle. On the unit circle, for any angle \\(\theta\), the tangent is given by:\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]
There are several key characteristics of the tangent function:

• Tangent values repeat every 180°, which means its periodicity is \(\pi\) radians, unlike sine and cosine, which repeat every 360°.
• In the fourth quadrant, the tangent function is negative because cosine is positive and sine is negative here.
• For 300°, using the reference angle of 60° and understanding the sign, we find \(\tan(300°) = -\sqrt{3}\).

Mastering the tangent function provides insight into real-world situations involving slopes and angles.

Quadrant System

The Quadrant System, a major component of the coordinate geometry, assists in analyzing the sign and value of trigonometric functions based on their position. The Cartesian plane is divided into four quadrants:

• Quadrant I: angles from 0° to 90°, all trigonometric functions are positive.
• Quadrant II: angles from 90° to 180°, sine is positive, cosine and tangent are negative.
• Quadrant III: angles from 180° to 270°, tangent is positive, sine and cosine are negative.
• Quadrant IV: angles from 270° to 360°, cosine is positive, sine and tangent are negative.

Understanding the quadrant system helps you quickly determine the sign of a trigonometric function based on the angle's position.

In this exercise, knowing that 300° falls in the fourth quadrant instantly tells you that the tangent will be negative. This knowledge speeds up the process of solving trigonometric problems and verifies the correctness of calculations.