Problem 16 - Algebra 2

Question

Find the exact value of each function. \(\sin \left(\frac{14 \pi}{6}\right)\)

Step-by-Step Solution

Answer

\(\frac{\sqrt{3}}{2}\)

Step 1: Simplify the Angle

First, simplify the angle \(\frac{14\pi}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives \(\frac{14}{6} = \frac{7}{3}\), hence \(\frac{14\pi}{6} = \frac{7\pi}{3}\).

Step 2: Find an Equivalent Angle in the Unit Circle

The angle \(\frac{7\pi}{3}\) is greater than \(2\pi\). Subtract \(2\pi\) to find an equivalent angle within the range of \([0, 2\pi)\). \(\frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3}\).

Step 3: Solve Using the Unit Circle

Now that the equivalent angle is \(\frac{\pi}{3}\), use the unit circle to find \(\sin\left(\frac{\pi}{3}\right)\). From the unit circle, \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\).

Key Concepts

Unit Circle

Understanding the unit circle is key to mastering trigonometry. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is crucial for finding trigonometric functions, including sine, cosine, and tangent, and the circle's special angles make calculations easier.

• The circumference of the unit circle is used as a reference to define angles in both degrees and radians.
• Points on the unit circle are identified by their coordinates \((\cos(\theta), \sin(\theta))\).
• With the unit circle, you can also see symmetries that help simplify trigonometric calculations.

Mastery of the unit circle means knowing its key angles and the values of sine and cosine at these angles. Always practice referring to the unit circle to quickly determine trigonometric ratios for different angles.

Sine Function

The sine function is a fundamental concept in trigonometry connecting the angles of a triangle to the ratio of the length of the opposite side over hypotenuse in a right-angled triangle. On the unit circle, the sine of an angle \(\theta\) is the y-coordinate of the point where the angle intersects the circle.

• Important for calculating wave patterns, oscillations, and more.
• Periodicity: The sine function repeats every \(2\pi\) radians.
• Range: The values of the sine function are from -1 to 1.
• Key angles: Knowing values like \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\) and \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\) is essential.

Understanding the sine function on the unit circle will improve your comprehension of angle and triangle relationships.

Angle Simplification

Simplifying angles is a common task when working with trigonometric functions. In our example, the angle \(\frac{14\pi}{6}\) was simplified to \(\frac{7\pi}{3}\) by dividing by the greatest common divisor.

• Crucial skill: Helps in identifying angles that can be simplified or normalized.
• Once reduced, check if the angle is greater than \(2\pi\).
• If necessary, find an equivalent angle by subtracting \(2\pi\) to bring it within one full circle.

Always simplify angles when possible to make calculations more manageable and connect them back to familiar values on the unit circle.

Radians Conversion

Radians are a natural way to measure angles as they relate directly to the circle's radius. Converting angles from degrees to radians, or handling angles in radians, is essential for trigonometry.

• Formula: Use \(\pi\) radians for 180 degrees. This means \(1\) radian = \(\frac{180}{\pi}\) degrees.
• Faster calculations: In radians, many trigonometric calculations are simpler and more direct.
• Common conversions include multiples like \(\frac{\pi}{2}, \pi, \frac{3\pi}{2},\) and \(2\pi\).

Frequently practice converting between degrees and radians to become comfortable with both. This versatility is indispensable for solving trigonometric problems efficiently.