Use the unit circle shown to find the value of the trigonometric function.
tan 11𝜋/6

Verified step by step guidance

1. Step 1: Understand that the unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. The angle 11𝜋/6 is measured in radians.
2. Step 2: Convert the angle 11𝜋/6 to a more familiar angle by subtracting 2𝜋 (a full circle) from it, since 11𝜋/6 is greater than 2𝜋. This will help find the equivalent angle within the first circle rotation.
3. Step 3: Calculate 11𝜋/6 - 2𝜋. Since 2𝜋 is equivalent to 12𝜋/6, subtracting gives 11𝜋/6 - 12𝜋/6 = -𝜋/6. This means 11𝜋/6 is coterminal with -𝜋/6.
4. Step 4: Recognize that -𝜋/6 is equivalent to 11𝜋/6 on the unit circle, and it corresponds to the reference angle 𝜋/6 in the fourth quadrant.
5. Step 5: Use the fact that in the fourth quadrant, the tangent function is negative. The reference angle 𝜋/6 has a tangent value of 1/√3, so tan(11𝜋/6) = -1/√3.

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it allows for the definition of trigonometric functions based on angles measured from the positive x-axis. Each point on the unit circle corresponds to a specific angle and its sine and cosine values, which are essential for calculating other trigonometric functions.

Tangent Function

The tangent function, denoted as tan(θ), is defined as the ratio of the sine and cosine of an angle: tan(θ) = sin(θ) / cos(θ). It represents the slope of the line formed by the angle in the unit circle. Understanding how to derive the tangent from the unit circle is crucial for solving problems involving angles and their corresponding trigonometric values.

Angle Measurement in Radians

In trigonometry, angles can be measured in degrees or radians, with radians being the standard unit in mathematical contexts. The angle 11π/6 radians corresponds to 330 degrees, which is important for locating the angle on the unit circle. Recognizing how to convert between radians and degrees is essential for accurately determining the values of trigonometric functions.