Problem 65 - Precalculus

Question

Explain the mistake that is made. Use the unit circle to evaluate \(\tan \left(\frac{5 \pi}{6}\right)\) exactly. \(\begin{array}{l}\text { Tangent is the } \\ \text { ratio of sine } \\\ \text { to cosine. }\end{array} \tan \left(\frac{5 \pi}{6}\right)=\frac{\sin \left(\frac{5 \pi}{6}\right)}{\cos \left(\frac{5 \pi}{6}\right)}\) Use the unit circle to \(\begin{array}{l}\text { identify sine } \\ \text { and cosine. }\end{array} \quad \sin \left(\frac{5 \pi}{6}\right)=-\frac{\sqrt{3}}{2}\) and \(\cos \left(\frac{5 \pi}{6}\right)=\frac{1}{2}\) Substitute values for \(\begin{array}{l}\text { sine and } \\ \text { cosine. }\end{array} \quad \tan \left(\frac{5 \pi}{6}\right)=\frac{-(\sqrt{3} / 2)}{1 / 2}\) Simplify. \(\tan \left(\frac{5 \pi}{6}\right)=-\sqrt{3}\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was incorrect signs for sine and cosine in the second quadrant.

Step 1: Recall the Unit Circle

The unit circle helps us find the sine and cosine of angles. For the angle \(\frac{5 \pi}{6}\), which lies in the second quadrant, we use the properties of the reference angle and signs in the second quadrant.

Step 2: Reference Angle and Signs

The reference angle for \(\frac{5 \pi}{6}\) is \(\pi/6\). In the second quadrant, sine is positive and cosine is negative. Thus, we have: \(\sin(\frac{5\pi}{6}) = \frac{1}{2}\) and \(\cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}\).

Step 3: Define Tangent

Tangent of an angle is the ratio of sine to cosine: \(\tan\theta = \frac{\sin\theta}{\cos\theta}\). Therefore, it's important to use the correct values for sine and cosine from the unit circle.

Step 4: Calculate \(\tan(\frac{5\pi}{6})\)

Substitute the correct values into the tangent formula: \(\tan\left(\frac{5\pi}{6}\right) = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}\). Simplifying this gives \(-\frac{1}{\sqrt{3}}\), which further simplifies to \(-\frac{\sqrt{3}}{3}\).

Step 5: Identify the Mistake

The mistake was in assigning the incorrect signs to sine and cosine. The sine in the second quadrant should be positive, and cosine should be negative, not the other way around.

Key Concepts

Trigonometric Functions

Trigonometric functions are a fundamental aspect of mathematics, especially when dealing with angles and circles. These functions help us understand relationships in triangles and the unit circle.

• Sine ( \( \sin \theta \) ): Measures the vertical distance from the center of the circle to the arc described by the angle \( \theta \) on the unit circle.
• Cosine ( \( \cos \theta \) ): Measures the horizontal distance from the center of the circle to the arc described by the angle \( \theta \) on the unit circle.
• Tangent ( \( \tan \theta \) ): Represents the ratio of the sine to the cosine, which is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

In solving trigonometric problems, it is crucial to use accurate values for sine and cosine, particularly when encountered on the unit circle.

Tangent

The tangent of an angle is one of the primary trigonometric functions and is denoted as \( \tan \theta \). This function can be visualized as the slope of the line that intersects the unit circle with angle \( \theta \).

• Tangent is formulated as the ratio of sine and cosine.
• Tangent can be positive or negative depending on the quadrant of the angle.

In any mathematical problem involving tangent, determining the correct values for sine and cosine is essential. For instance, for \( \tan\left(\frac{5\pi}{6}\right) \), correct calculation involves their values in the second quadrant, where tangent is negative.

Reference Angle

In trigonometry, the reference angle is always a positive acute angle that helps in finding the trigonometric values of the original angle, regardless of which quadrant it lies in.

• The reference angle \( \theta_{ref} \) is always formed with the x-axis.
• In the unit circle, the reference angle becomes crucial when evaluating trigonometric functions for angles located in quadrants other than the first.

For the problem with \( \frac{5\pi}{6} \), our reference angle is \( \frac{\pi}{6} \), which helps to determine the signs and values of sine and cosine for computing tangent accurately.

Second Quadrant

In the unit circle, the second quadrant is significant as it impacts the sign of trigonometric functions. It covers angles between \( \frac{\pi}{2} \) and \( \pi \).

• Sine: In this quadrant, the sine of an angle is positive as the y-values (vertical) are above the x-axis.
• Cosine: The cosine of an angle is negative since the x-values (horizontal) are to the left of the origin.
• Tangent, as a consequence, will be negative because it depends on the ratio of sine over cosine.

Careful attention must be paid to the signs of sine and cosine when computing tangent in this quadrant. This was the critical error in the initial problem which led to the incorrect solution.