Problem 34 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \sin \frac{11 \pi}{6} $$

Step-by-Step Solution

Answer

The exact value of \(\sin \frac{11\pi}{6}\) is \(-\frac{1}{2}\).

Step 1: Convert the Angle to Degrees

First, convert the given angle from radians to degrees. The formula to convert radians to degrees is: \[\text{Degrees} = \frac{\text{Radians} \times 180}{\pi}\]Substituting \(\text{Radians} = \frac{11\pi}{6}\), we have:\[\text{Degrees} = \frac{\frac{11\pi}{6} \times 180}{\pi} = 330°\]

Step 2: Identify the Reference Angle

The reference angle for an angle \(330°\) is the acute angle it forms with the x-axis. To find it, subtract the angle from \(360°\): \[360° - 330° = 30°\]Thus, the reference angle is \(30°\).

Step 3: Determine the Quadrant

Angle \(330°\) is in the fourth quadrant, where the sine function is negative. This information will help determine the sign of the sine value.

Step 4: Find the Sine of the Reference Angle

The sine of the reference angle \(30°\) is well-known: \[\sin 30° = \frac{1}{2}\]

Step 5: Apply the Sign

Since the angle \(330°\) is in the fourth quadrant where sine is negative, apply the sign to the sine of the reference angle: \[\sin 330° = -\frac{1}{2}\]

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It relates an angle in a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse. This is expressed as \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \). The sine function is periodic with a period of \(2\pi\) radians or \(360^\circ\).

In the context of the coordinate plane, the sine of an angle corresponds to the y-coordinate of a point on the unit circle. For instance, at \(0^\circ\), \( \sin \theta = 0 \), and it achieves its maximum value of \(1\) at \(90^\circ\) (or \(\frac{\pi}{2}\) radians).

• The sine function moves from \(0\) to \(1\) as the angle increases from \(0^\circ\) to \(90^\circ\).
• Between \(90^\circ\) and \(180^\circ\), it decreases back to \(0\).
• As the angle continues through \(180^\circ\) to \(270^\circ\), the sine becomes negative, reaching \(-1\).
• Finally, it returns to \(0\) at \(360^\circ\).

Understanding this oscillatory movement is crucial for interpreting the sine function's value at any given angle.

Reference Angle

A reference angle is the smallest angle between the terminal side of a given angle and the x-axis. It helps simplify the calculations of trigonometric functions by reducing a given angle to the first quadrant, where all functions have their basic positive values.

For example, for an angle of \(330^\circ\) in the fourth quadrant, the reference angle can be calculated by finding the difference between \(360^\circ\) and \(330^\circ\), which is \(30^\circ\).

• This simplifies the process because you can use the familiar acute angle values to find trigonometric functions.
• The values are then adjusted according to the original angle's quadrant to determine the sign (positive or negative).

Understanding reference angles is a powerful shortcut, especially when dealing with non-acute angles in trigonometry.

Fourth Quadrant

The coordinate plane is divided into four quadrants, and the fourth quadrant is the section where both the x-value is positive, and the y-value is negative. It is crucial to understand the sign behavior of trigonometric functions in the fourth quadrant.

Parameters in the fourth quadrant range from \(270^\circ\) to \(360^\circ\) or equivalent radian measures from \(\frac{3\pi}{2}\) to \(2\pi\).

• In this quadrant, the cosine function is positive, while the sine and tangent functions are negative.
• For example, \(\sin 330^\circ = -\frac{1}{2}\) because \(330^\circ\) lies in this quadrant.

Knowing which trigonometric functions are positive or negative in each quadrant allows for quick analysis and solutions of problems involving angles larger than \(90^\circ\).