Problem 377 - Precalculus

Question

Use reference angles to evaluate \(\tan 210^{\circ}\)

Step-by-Step Solution

Answer

The value of \( \tan 210^{\circ} \) is \( \frac{\sqrt{3}}{3} \).

Step 1: Determine the Related Reference Angle

A reference angle is the acute angle a given angle makes with the x-axis. For angles over 180°, find the reference angle by subtracting 180° from the given angle. In this case, the reference angle for 210° is computed as follows: 210° - 180° = 30°.

Step 2: Identify the Quadrant for the Given Angle

The angle 210° is in the third quadrant of the unit circle, where both sine and cosine values are negative. The tangent function is defined as the ratio of sine to cosine, which is positive in the third quadrant.

Step 3: Find the Tangent of the Reference Angle

The tangent of the reference angle 30° is known from trigonometry: \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \) or \( \tan 30^{\circ} = \frac{\sqrt{3}}{3} \) after rationalizing the denominator. However, the tangent in the third quadrant is negative, so \( \tan 210^{\circ} = \tan 30^{\circ} = \frac{1}{\sqrt{3}} \).

Step 4: Apply the Sign Related to the Quadrant

Since the tangent value should be positive in the third quadrant (as both sine and cosine values are negative, resulting in a positive ratio), the value of \( \tan 210^{\circ} = \frac{1}{\sqrt{3}} \) should be positive, but upon re-evaluation we realize it should match \( \tan 210^{\circ} = \frac{\sqrt{3}}{3} \). Therefore, assess and confirm using trigonometric identities.

Key Concepts

Reference Angles

To begin with, understanding reference angles is key to working with trigonometric functions. A reference angle is the smallest positive angle that a given angle makes with the x-axis. It is always measured in degrees between 0° and 90°, making it an acute angle.
This concept simplifies the calculation of trigonometric functions for angles larger than 90° and smaller than -90°. When evaluating an angle like 210°, you need to simplify it by determining its reference angle. For angles between 180° and 360°, the reference angle is found by subtracting 180° from the angle. For 210°, the calculation is simple:

• Reference angle = 210° - 180° = 30°.

Reference angles help by connecting given angles to familiar acute angles, making it easier to find sine, cosine, and tangent values.

Unit Circle

The unit circle is a fundamental tool in trigonometry used to define and understand trigonometric functions. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. The angle measures are mapped on the circle, with common angles like 30°, 45°, and 60° producing standard trigonometric values. These angles, plotted as points on the circle, relate the angle's measure to positions on the coordinate plane.

• In the third quadrant, both sine and cosine are negative.
• The tangent function, as a ratio of sine to cosine, will be positive since it involves dividing two negative numbers.

Thus, understanding how angles are classified in quadrants on the unit circle is crucial for accurately calculating trigonometric functions.

Tangent Function

The tangent function is one of the primary trigonometric functions, symbolized as \(\tan\). It is defined mathematically as the ratio of the opposite side to the adjacent side in a right triangle, or simply as sine divided by cosine: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).For the angle 210°, once the reference angle is identified as 30°, you can use known trigonometric values:

• \(\tan(30°) = \frac{1}{\sqrt{3}}\), which simplifies to \(\frac{\sqrt{3}}{3}\) when rationalized.

Since the angle is in the third quadrant, a positive tangent value is expected:

• This occurs because both sine and cosine are negative here, giving a positive ratio when divided.

Therefore, \(\tan 210^{\circ} = \frac{\sqrt{3}}{3}\). Grasping this concept allows you to solve trigonometric problems involving tangent with ease.