Use the unit circle shown to find the value of the trigonometric function. tan 11π/6
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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.
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.
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.
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.
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/β3, so tan(11π/6) = -1/β3.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
The tangent function, denoted as tan(ΞΈ), is defined as the ratio of the sine and cosine of an angle: tan(ΞΈ) = sin(ΞΈ) / cos(ΞΈ). 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.
In trigonometry, angles can be measured in degrees or radians, with radians being the standard unit in mathematical contexts. The angle 11Ο/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.