Problem 95 - Precalculus

Question

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

Step-by-Step Solution

Answer

\( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \)

Step 1: Understand the Angle

The exercise asks for the exact value of \( \cos \frac{\pi}{6} \). The angle \( \frac{\pi}{6} \) is in radians and corresponds to 30 degrees in the unit circle.

Step 2: Recall Cosine of Special Angles

On the unit circle, certain angles have known cosine and sine values. The angle \( 30^{\circ} \) or \( \frac{\pi}{6} \) is one of these special angles. The cosine of \( 30^{\circ} \) is a known value.

Step 3: Find the Cosine Value

From trigonometric knowledge, we know that \( \cos 30^{\circ} = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \). This is derived from the unit circle values.

Key Concepts

Unit Circle

The unit circle is a fundamental concept when dealing with trigonometric functions. It's a circle with a radius of 1, centered at the origin of a coordinate plane. Understanding the unit circle helps in comprehending various trigonometric functions, including sine, cosine, and tangent.

On this circle:

• The horizontal axis represents the cosine values, while the vertical represents the sine values.
• When you rotate around the unit circle, the angle usually corresponds to the amount of rotation from the positive x-axis.
• These angles can be represented in radians or degrees.

This is a valuable tool because it translates circular motion into linear values, helping to solve trigonometric problems easily. Knowing this can simplify how we find exact trigonometric values.

Special Angles

Special angles are specific angles on the unit circle where the sine, cosine, and tangent values are easily recognizable and memorable. These include angles like 0°, 30°, 45°, 60°, and 90°, or their equivalent in radians, like \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \) and \( \frac{\pi}{2} \).

For these angles, you should practice and remember their trigonometric values because:

• They frequently appear in mathematical problems.
• The values are derived from the geometry of the 30-60-90 and 45-45-90 triangles.
• Knowing these can speed up solving trigonometric problems significantly.

For example, for a 30° angle or \( \frac{\pi}{6} \) radians, the cosine value is \( \frac{\sqrt{3}}{2} \), and this becomes a handy fact when calculating trigonometric values without a calculator.

Cosine

The cosine function measures the horizontal distance of a point on the unit circle from the origin. This function is essential as it helps determine the cosine values for angles measuring circular rotations.

Key points about cosine:

• The cosine of an angle can be thought of as the x-coordinate of an endpoint on the unit circle.
• For special angles, such as \( \frac{\pi}{6} \) or 30°, the cosine value is predetermined and equals \( \frac{\sqrt{3}}{2} \).
• Cosine values cycle in a periodic manner with a period of \(2\pi \), which means the values repeat after every full rotation.

This periodic nature makes cosine a powerful function in modeling waves and vibrations in engineering and sciences. Understanding cosine through the unit circle and special angles can greatly enhance your ability to quickly and correctly solve trigonometric equations.