In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator.csc 7𝜋6

Verified step by step guidance

1. Identify the angle: The given angle is \( \frac{7\pi}{6} \).
2. Determine the reference angle: Since \( \frac{7\pi}{6} \) is in the third quadrant, subtract \( \pi \) from \( \frac{7\pi}{6} \) to find the reference angle. The reference angle is \( \frac{7\pi}{6} - \pi = \frac{\pi}{6} \).
3. Recall the cosecant function: \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
4. Use the reference angle: The sine of the reference angle \( \frac{\pi}{6} \) is \( \frac{1}{2} \). Therefore, \( \csc(\frac{\pi}{6}) = 2 \).
5. Determine the sign: Since \( \frac{7\pi}{6} \) is in the third quadrant where sine is negative, \( \csc(\frac{7\pi}{6}) = -2 \).

Key Concepts

Reference Angles

A reference angle is the acute angle formed by the terminal side of an angle in standard position and the x-axis. It is always positive and is used to simplify the calculation of trigonometric functions. For angles greater than 360 degrees or less than 0 degrees, the reference angle is found by reducing the angle to its equivalent within the first rotation (0 to 2π radians).

Cosecant Function

The cosecant function, denoted as csc, is the reciprocal of the sine function. It is defined as csc(θ) = 1/sin(θ). To find the exact value of csc(7π/6), one must first determine the sine of the angle, which can be derived from its reference angle and the quadrant in which the angle lies.

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 visualization of angles and the values of trigonometric functions. The coordinates of points on the unit circle correspond to the cosine and sine values of the angles, facilitating the calculation of trigonometric functions like cosecant.