Problem 90 - Precalculus

Question

For the following exercises, find the exact value of each trigonometric function. $$ \sin \frac{\pi}{6} $$

Step-by-Step Solution

Answer

The exact value of \( \sin \frac{\pi}{6} \) is \( \frac{1}{2} \).

Step 1: Understand the Problem

We need to find the exact value of the sine function for the angle \( \frac{\pi}{6} \) radians.

Step 2: Recall the Special Angle Identity

\( \frac{\pi}{6} \) radians is equivalent to 30 degrees. From trigonometric identities, we know that \( \sin 30^\circ = \frac{1}{2} \).

Step 3: Convert the Identity to Radians

Since \( 30^\circ \) is equivalent to \( \frac{\pi}{6} \) radians, we use the identity \( \sin 30^\circ = \frac{1}{2} \) to find that \( \sin \frac{\pi}{6} = \frac{1}{2} \).

Key Concepts

Exact Values of Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent can often be expressed with exact values for certain "special" angles. These exact values are frequently used in math problems and have been standardized for key angles because they often simplify calculations. For instance, when you need to find the trigonometric value of an angle like \( \frac{\pi}{6} \) (commonly known as 30 degrees), you can derive its exact value from established trigonometric identities. Knowing the exact values:

• Helps solve equations more efficiently.
• Reduces the chance of errors that come from approximation.

By using known exact values, like \( \sin \frac{\pi}{6} = \frac{1}{2} \), learners can confidently move forward in solving more complex problems without relying on a calculator.

Special Angle Identities

Special angle identities are a cornerstone of trigonometry, providing exact values for trigonometric functions at key angles like 0°, 30°, 45°, 60°, and 90°. These angles correspond to radian measures of 0, \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{2} \) respectively.Key exact values:

• \( \sin 30^\circ = \frac{1}{2} \)
• \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
• \( \tan 30^\circ = \frac{\sqrt{3}}{3} \)

Special angle identities provide a quick reference since they are derived from the unit circle and basic geometric relations. Understanding these identities allows problem solvers to anticipate and recognize patterns in complex trigonometric problems, smoothing the path to finding solutions with greater ease.

Radian to Degree Conversion

Angles can be expressed in two ways: degrees and radians. It’s important to be comfortable converting between these two since some problems will present angles in one format while the known identities or solutions are often in the other. The conversion between degrees and radians is straightforward:

• \(180^\circ = \pi \text{ radians}\)
• To convert from radians to degrees, multiply by \(\frac{180}{\pi}\)
• Conversely, from degrees to radians, multiply by \(\frac{\pi}{180}\)

In our specific exercise, converting \( \frac{\pi}{6} \) radians to degrees involves multiplying \( \frac{\pi}{6} \times \frac{180}{\pi} \), which yields 30 degrees. This allows one to quickly find the sine of 30 degrees using the special angle identities, confirming \( \sin \frac{\pi}{6} = \frac{1}{2} \). Mastering radian to degree conversion ensures that you can tackle trigonometric questions presented in any format.