Problem 6 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{5 \pi}{6}\) (b) \(\cos \frac{5 \pi}{6}\) (c) \(\tan \frac{5 \pi}{6}\)

Step-by-Step Solution

Answer

(a) \( \frac{1}{2} \), (b) \( -\frac{\sqrt{3}}{2} \), (c) \( -\frac{\sqrt{3}}{3} \)

Step 1: Identify the Reference Angle

The angle \( \frac{5 \pi}{6} \) is in the unit circle. The reference angle is \( \frac{\pi}{6} \), also known as 30 degrees, because \( \frac{5\pi}{6} \) is 30 degrees away from \( \pi \), or 180 degrees.

Step 2: Determine the Quadrant

The angle \( \frac{5 \pi}{6} \) is in the second quadrant because it is less than \( \pi \) (or 180 degrees) but greater than \( \frac{\pi}{2} \) (or 90 degrees).

Step 3: Evaluate \( \sin \frac{5 \pi}{6} \)

In the second quadrant, the sine function is positive. \( \sin \frac{\pi}{6} = \frac{1}{2} \), so \( \sin \frac{5\pi}{6} = \frac{1}{2} \).

Step 4: Evaluate \( \cos \frac{5 \pi}{6} \)

In the second quadrant, the cosine function is negative. \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \), so \( \cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2} \).

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

The tangent function is the ratio of sine to cosine. So, \( \tan \frac{5\pi}{6} = \frac{\sin \frac{5\pi}{6}}{\cos \frac{5\pi}{6}} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \) after rationalizing the denominator.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It simplifies understanding trigonometric functions by relating them to angles on a circle. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. All trigonometric functions can be visualized using this circle.
The unit circle is divided into four quadrants, corresponding to the four sections formed by the x and y axes. Angles in trigonometry are typically measured in radians, where the full circle is equal to \(2\pi\) radians (or 360 degrees). If you imagine moving counterclockwise from the positive x-axis, each point on the circle corresponds to a unique angle.

• The rightmost point on the circle at the x-axis is \(0\) radians or \(2\pi\).
• The topmost point at the y-axis is \(\frac{\pi}{2}\).
• The leftmost point at the x-axis is \(\pi\).
• The bottommost point at the y-axis is \(\frac{3\pi}{2}\).

The coordinates of any point on the unit circle are \((\cos(\theta), \sin(\theta))\). This means for any angle \(\theta\), the x-coordinate gives the cosine of \(\theta\), and the y-coordinate gives the sine of \(\theta\). Utilizing the unit circle helps identify the trigonometric values easily.

Reference Angle

A reference angle is a helpful concept to simplify the calculation of trigonometric functions. It is the smallest angle that a terminal side makes with the x-axis. The reference angle is always between \(0\) and \(\frac{\pi}{2}\) or \(0\) and 90 degrees.
To find the reference angle for a given angle in any quadrant:

• If the angle is in the first quadrant, the reference angle is the angle itself.
• In the second quadrant, subtract the angle from \(\pi\) (180 degrees).
• In the third quadrant, subtract \(\pi\) from the angle.
• In the fourth quadrant, subtract the angle from \(2\pi\).

In the case of \(\frac{5\pi}{6}\), since it is in the second quadrant, we calculate the reference angle by \(\pi - \frac{5\pi}{6}\), which simplifies to \(\frac{\pi}{6}\). This is also known as 30 degrees. Recognizing the reference angle aids in determining the sine, cosine, and tangent of the original angle. The trigonometric functions of the reference angle are frequently used to find corresponding values for the original angle.

Quadrants in Trigonometry

Understanding the quadrants on the unit circle is crucial to learning trigonometric functions. Each quadrant offers specific sign patterns for different trigonometric functions.
Quadrants in the unit circle are numbered counterclockwise:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, while cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, but cosine is positive.

Knowing which quadrant an angle is in helps determine the sign for each trigonometric function.
For example, the angle \(\frac{5\pi}{6}\) lies in the second quadrant. Here, as per the rules:

• \(\sin(\frac{5\pi}{6})\) is positive because sine is positive in the second quadrant.
• \(\cos(\frac{5\pi}{6})\) is negative because cosine is negative in the second quadrant.
• The tangent \((\tan)\) value, which is the ratio of sine to cosine, becomes negative.

These signs and their quadrant-specific behaviors simplify the process of calculating trigonometric functions for given angles.