Problem 370 - Precalculus

Question

Find the exact value of \(\sin \frac{\pi}{6}\)

Step-by-Step Solution

Answer

\( \sin \frac{\pi}{6} = \frac{1}{2} \).

Step 1: Understanding the Problem

We need to find the exact value of the sine function for the angle \( \frac{\pi}{6} \). Knowing that angles in radians are more often used in trigonometry, \( \frac{\pi}{6} \) radians is one of the angles whose sine value is commonly memorized.

Step 2: Convert to Degrees (Optional)

To help visualize the angle, we convert \( \frac{\pi}{6} \) to degrees. We know that \( \pi \) radians equals 180 degrees, so \( \frac{\pi}{6} \) is equivalent to 30 degrees. This isn't necessary for solving but can aid in understanding.

Step 3: Recall Sine Value for 30 Degrees

The sine of commonly-used angles like 30 degrees (or \( \frac{\pi}{6} \) radians) is usually memorized. For a 30-degree angle, \( \sin 30^{\circ} = \frac{1}{2} \).

Step 4: Use Trigonometric Identities (Confirmed Knowledge)

To verify, use a special triangle, such as a 30-60-90 triangle. In a 30-60-90 triangle, the sides are in the ratio 1 : \( \sqrt{3} \) : 2. The sine of 30 degrees, which is the ratio of the opposite side to the hypotenuse (1/2), confirms \( \sin \frac{\pi}{6} = \frac{1}{2} \).

Step 5: Conclusion

Therefore, based on the unit circle and triangle reasoning, the exact value of \( \sin \frac{\pi}{6} \) is \( \frac{1}{2} \).

Key Concepts

Understanding Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and lengths of triangles. It focuses mainly on right-angled triangles, where one of the angles is always 90 degrees. Trigonometric functions like sine, cosine, and tangent help us relate the angles to side lengths of these triangles. Trigonometry is essential in many fields like physics, engineering, and even music.

• The sine function, represented as \( \sin \), is one of the primary trigonometric functions. It calculates the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle.
• Understanding trigonometry requires memorization of sine, cosine, and tangent values for commonly used angles like 30°, 45°, and 60° (or their radian equivalents \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \)).

By mastering these fundamental angles and their sine values, students can simplify solving trigonometric problems.

Radians to Degrees Conversion

Sometimes, angles are given in radians instead of degrees, especially in higher-level mathematics and physics. Radians offer a measure of angle based on the radius of a circle. The conversion between radians and degrees is crucial for understanding because degrees are more intuitive for many people.
To convert radians to degrees, use the conversion factor \( 180/\pi \):

• Multiply the radian value by \( 180/\pi \) to get the equivalent in degrees.
• For instance, converting \( \frac{\pi}{6} \) radians into degrees involves calculating \( \frac{\pi}{6} \times \frac{180}{\pi} = 30 \) degrees.

This conversion is a helpful tool in trigonometry, providing context and making it easier for those who find angles in degrees more manageable. Therefore, understanding both systems and how to convert between them is beneficial.

Special Triangles and Their Properties

In trigonometry, special triangles provide a useful shortcut for finding exact values of trigonometric functions without a calculator. The most notable special triangles are the 30-60-90 and 45-45-90 triangles.
In a 30-60-90 triangle:

• The side opposite the 30-degree angle is half the hypotenuse.
• This results in sine, cosine, and tangent values that are often memorized: \( \sin 30^{\circ} = \frac{1}{2} \), \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), \( \tan 30^{\circ} = \frac{\sqrt{3}}{3} \).

The 45-45-90 triangle, meanwhile, helps with angles that are 45 degrees:

• This triangle has sides in the ratio 1:1:√2, which means \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \).

Understanding these special triangles reduces the need for a calculator and provides a visual understanding of trigonometric concepts.

The Unit Circle and Its Importance

The unit circle is a fundamental concept in trigonometry, providing a comprehensive way to understand trigonometric functions in terms of angle measures. This circle has a radius of 1 and is centered at the origin of a coordinate plane.
Here's how the unit circle helps in understanding trig functions:

• Angles are measured in radians as the length of an arc on this circle.
• The x-coordinate of a point on the unit circle represents the cosine of the angle, and the y-coordinate represents the sine.
• Knowing the points of intersections of the circle with the axes and special angles, like 30°, 45°, and 60°, allows for quick calculations of trig functions.

For instance, for \( \frac{\pi}{6} \) radians (30 degrees), the coordinates are \( (\frac{\sqrt{3}}{2}, \frac{1}{2}) \), confirming \( \sin \frac{\pi}{6} = \frac{1}{2} \). By learning and utilizing the unit circle, students can gain a strong grasp of trigonometric values across different angles.