Problem 61 - Precalculus

Question

Evaluate the following expressions exactly: $$\tan 210^{\circ}$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Quadrant

The given angle is \(210^\circ\). To find which quadrant this lies in, recall that an angle in standard position is measured counterclockwise from the positive x-axis. Since one full rotation is \(360^\circ\), an angle of \(210^\circ\) is greater than \(180^\circ\) and less than \(270^\circ\). Hence, it lies in the third quadrant.

Step 2: Reference Angle Calculation

In the third quadrant, the reference angle is the angle formed with the negative x-axis. To find the reference angle of \(210^\circ\), subtract \(180^\circ\) from \(210^\circ\):\[ 210^\circ - 180^\circ = 30^\circ \].Thus, the reference angle is \(30^\circ\).

Step 3: Evaluate the Tangent Function

The tangent function in the third quadrant is positive. Since the reference angle is \(30^\circ\), use the identity \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \). As the third quadrant makes the tangent positive, \(\tan(210^\circ) = \tan(30^\circ) = \frac{1}{\sqrt{3}}\). It's often rationalized to \(\frac{\sqrt{3}}{3}\).

Key Concepts

Understanding the Tangent Function

The tangent function is one of the fundamental trigonometric functions alongside sine and cosine. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. In terms of angles on the unit circle, the tangent of an angle \(\theta\) is calculated as the sine of \(\theta\) divided by the cosine of \(\theta\), i.e., \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}. \]This definition makes it clear that the function is undefined when the cosine of \(\theta\) is zero, as division by zero is undefined.Working with tangent functions on the unit circle can help understand their behavior over different quadrants. The tangent function repeats every \(180^\circ\) because its fundamental period is one complete turn around the circle.

• If the angle is in the first quadrant, both sine and cosine are positive, so tangent is positive.
• In the third quadrant, both sine and cosine are negative, making the tangent positive as well.
• In the second and fourth quadrants, the signs of sine and cosine differ, leading to a negative tangent.

These points help determine whether the value of the tangent function is positive or negative depending on the quadrant.

Determining a Reference Angle

When working with angles, especially those that are not standard angles like \(30^\circ, 45^\circ,\) or \(60^\circ\), determining a reference angle is a useful tool.A reference angle is always a positive acute angle, measuring between \(0^\circ\) and \(90^\circ\), which represents the smallest angle between the terminal side of the given angle and the x-axis.This simplification allows you to relate any angle to a known trigonometric value, especially on a unit circle.Calculating a reference angle depends on the quadrant in which the original angle resides:

• In the first quadrant, the reference angle is the same as the angle.
• In the second quadrant, subtract the angle from \(180^\circ\).
• In the third quadrant, subtract \(180^\circ\) from the angle.
• In the fourth quadrant, subtract the angle from \(360^\circ\).

Determining the reference angle for a trigonometric function such as tangent helps find precise values quickly and accurately, matched to integers or simple fractions.

Executing Quadrant Analysis

Quadrant analysis is essential in understanding how trigonometric functions behave differently based on the angle's positionon the coordinate plane. The coordinate system is divided into four quadrants that function as follows:

• Quadrant I: Here, all trigonometric functions are positive.
• Quadrant II: Sine is positive, while cosine and tangent are negative.
• Quadrant III: Tangent is positive, while sine and cosine are negative.
• Quadrant IV: Cosine is positive, but sine and tangent are negative.

To perform quadrant analysis, first determine which quadrant the given angle lies in. Since each quadrant has different behaviors of the trigonometric functions, analyzingwhich function remains positive or becomes negative is critical. For instance, if we analyze an angle of \(210^\circ\), we find it in the third quadrant.This tells us that the tangent of the angle is positive, guided by quadrant-specific knowledge that both sine and cosine are negative in this quadrant, resulting in a positive tangent. Understanding quadrant analysis allows solving trigonometric equations effectively and predicting the sign of a function's output without performing extensive calculations.