Problem 54 - Precalculus

Question

Use the unit circle to find all of the exact values of \(\theta\) that make the equation true in the indicated interval. $$\tan \theta=\frac{\sqrt{3}}{3}, 0 \leq \theta \leq 2 \pi$$

Step-by-Step Solution

Answer

The angles are \( \frac{\pi}{6} \) and \( \frac{7\pi}{6} \).

Step 1: Understanding the Tangent Value

We need to find angles on the unit circle where the tangent of the angle equals \( \frac{\sqrt{3}}{3} \). We know that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Step 2: Recognizing Known Tan Values

Recall that \( \tan \theta = \frac{\sqrt{3}}{3} \) corresponds to standard angles. In the unit circle, \( \tan \frac{\pi}{6} = \frac{\sqrt{3}}{3} \). Thus, \( \theta = \frac{\pi}{6} \) is one solution.

Step 3: Determine Other Quadrant Solutions

The tangent function is positive in the first and third quadrants. From Step 2, since \( \tan \theta = \frac{\sqrt{3}}{3} \) and returns to positive after passing through the second quadrant, the next solution will be \( \theta = \frac{\pi}{6} + \pi \), which is \( \theta = \frac{7\pi}{6} \).

Step 4: Confirming within Interval

Both \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{7\pi}{6} \) lie within the interval \( 0 \leq \theta \leq 2\pi \). No other solutions exist within this interval since the cycle of the tangent repeats every \( \pi \).

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions and often represented as \( \tan \theta \). Understanding the tangent function involves knowing its relation with sine and cosine. Specifically, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This equation highlights that tangent is essentially the ratio of the sine to the cosine of an angle.In the context of the unit circle, the tangent function is periodic, with a cycle repeating every \( \pi \) radians. This means that the values of tangent recur after the angle passes by \( 180^{\circ} \), or \( \pi \). The tangent is also an odd function, which means that \( \tan(-\theta) = -\tan(\theta) \). This property can help determine the behavior and values of tangent in various quadrants, further assisting in finding specific angles where a known tangent value appears.

Quadrant Solutions

When solving trigonometric equations using the unit circle, it's essential to consider the quadrant in which a solution lies. The unit circle is divided into four quadrants:

• Quadrant I: \(0 < \theta < \frac{\pi}{2}\)
• Quadrant II: \(\frac{\pi}{2} < \theta < \pi\)
• Quadrant III: \(\pi < \theta < \frac{3\pi}{2}\)
• Quadrant IV: \(\frac{3\pi}{2} < \theta < 2\pi\)

The tangent function is positive in the first and third quadrants. Hence, when looking for solutions to the equation \( \tan \theta = \frac{\sqrt{3}}{3} \), you would find them in these quadrants. For instance, knowing that \( \tan \theta = \frac{\sqrt{3}}{3} \) is true for \( \theta = \frac{\pi}{6} \) in Quadrant I, we can deduce that the other solution within the given interval would reside in Quadrant III at \( \theta = \frac{7\pi}{6} \). These solutions use both the knowledge of the value and the quadrant characteristics.

Standard Angles

Standard angles are specific angle measures on the unit circle that are often memorized because of their frequently occurring sine, cosine, and tangent values.For instance:- \( \tan \frac{\pi}{6} = \frac{\sqrt{3}}{3} \)- \( \tan \frac{\pi}{3} = \sqrt{3} \)- \( \tan \frac{\pi}{4} = 1 \)These angles are considered standard due to their simple, often rational trigonometric ratios. This makes them incredibly useful when solving trigonometric equations because they serve as reference points. In this exercise, recognizing \( \tan \theta = \frac{\sqrt{3}}{3} \) from one of these standard angles narrows down the potential solutions quickly. By zeroing in on such known values and leveraging their periodic nature, determining all solutions within a specific interval on the unit circle becomes more straightforward.