Problem 4 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin \frac{5 \pi}{6}} & {\text { (b) } \cos \frac{5 \pi}{6}} & {\text { (c) } \tan \frac{5 \pi}{6}}\end{array} $$

Step-by-Step Solution

Answer

\( \sin \frac{5\pi}{6} = \frac{1}{2}, \cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}, \tan \frac{5\pi}{6} = -\frac{\sqrt{3}}{3} \).

Step 1: Understanding the Angle

The angle \( \frac{5\pi}{6} \) is in radians. To understand its position on the unit circle, note that it corresponds to 150 degrees. This means it is in the second quadrant where sine is positive, and cosine and tangent are negative.

Step 2: Finding \( \sin \frac{5\pi}{6} \)

Since the angle \( \frac{5\pi}{6} \) is in the second quadrant, we use the reference angle \( \pi - \frac{5\pi}{6} = \frac{\pi}{6} \). The sine of \( \frac{5\pi}{6} \) is equal to the sine of \( \frac{\pi}{6} \), which is \( \frac{1}{2} \). So, \( \sin \frac{5\pi}{6} = \frac{1}{2} \).

Step 3: Finding \( \cos \frac{5\pi}{6} \)

In the second quadrant, cosine is negative. Using the reference angle \( \frac{\pi}{6} \), \( \cos \frac{5\pi}{6} = -\cos \frac{\pi}{6} \). The cosine of \( \frac{\pi}{6} \) is \( \frac{\sqrt{3}}{2} \), thus \( \cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2} \).

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

Tangent is the ratio of sine to cosine. Thus, \( \tan \frac{5\pi}{6} = \frac{\sin \frac{5\pi}{6}}{\cos \frac{5\pi}{6}} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} \). Simplifying gives \( \tan \frac{5\pi}{6} = -\frac{\sqrt{3}}{3} \).

Key Concepts

Unit Circle

The unit circle is a powerful tool for understanding trigonometric functions. It's a circle with a radius of 1 centered at the origin of a coordinate plane. Every point on the unit circle can be described using coordinates \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle from the positive x-axis.

• Angles on the unit circle can be measured in degrees or radians.
• The circle helps visualize the sign and values of trigonometric functions for different angles.
• Key angles, like the reference angles, simplify finding trigonometric values.

When you rotate around the unit circle, remember that the sine corresponds to the y-coordinate, while the cosine corresponds to the x-coordinate of the angle on the circle.

This makes it intuitive to determine when values will be positive or negative based on their quadrant.

Reference Angle

The reference angle is an angle made with the x-axis and is always positive and **between 0 and 90 degrees**.
It's used to find the values of sine and cosine for any angle. The reference angle helps simplify complex angles by converting them into an equivalent acute angle, allowing us to use known values from specific angles on the unit circle.

• For angles in the first quadrant, the reference angle is the angle itself.
• In the second quadrant, it is \(\pi - \theta\).
• In the third quadrant, it's \(\theta - \pi\).
• For the fourth quadrant, it is \(2\pi - \theta\).

Using the reference angle, you can leverage symmetry and properties of the unit circle to find exact trigonometric values across all four quadrants.

Quadrants

The coordinate plane is divided into four quadrants, each influencing the sign of trigonometric functions:

• **First Quadrant**: All trigonometric functions are positive.
• **Second Quadrant**: Sine is positive; cosine and tangent are negative.
• **Third Quadrant**: Tangent is positive; sine and cosine are negative.
• **Fourth Quadrant**: Cosine is positive; sine and tangent are negative.

Understanding which quadrant your angle resides in is crucial for determining the sign of sine, cosine, and tangent values.
For example, for the angle \(\frac{5\pi}{6}\), it's located in the second quadrant, therefore the sine is positive while cosine and tangent are negative.

Radian Measure

Radians are an alternative way to measure angles, used widely in trigonometry and calculus. One complete rotation around the circle is \(2\pi\) radians, equivalent to 360 degrees. This means:

• \(\pi\) radians is half a circle, or 180 degrees.
• \(\frac{\pi}{2}\) radians is a quarter circle, or 90 degrees.
• \(\frac{\pi}{4}\), \(\frac{\pi}{6}\), and \(\frac{3\pi}{2}\) are common fractional radian measures representing specific sections of the circle.

Radians provide a more natural connection between angle measures and arc length, since the length of the arc subtended by an angle in a unit circle is numerically equal to the radian measure of the angle.
This is why radians are preferred in advanced mathematics.