In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. csc(7𝜋/6)

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Identify the given angle: \(7\pi/6\). This angle is in radians and is greater than \(\pi\), so it lies in the third quadrant since \(\pi < 7\pi/6 < 3\pi/2\).

Find the reference angle for \(7\pi/6\). The reference angle \(\theta_r\) is the acute angle formed with the x-axis. For angles in the third quadrant, \(\theta_r = \theta - \pi\). So, calculate \(\theta_r = 7\pi/6 - \pi = \pi/6\).

Recall the definition of cosecant: \(\csc \theta = \frac{1}{\sin \theta}\). To find \(\csc 7\pi/6\), we need to find \(\sin 7\pi/6\) first.

Determine the sign of \(\sin 7\pi/6\). Since \(7\pi/6\) is in the third quadrant, where sine is negative, \(\sin 7\pi/6 = -\sin \pi/6\).

Use the known exact value \(\sin \pi/6 = \frac{1}{2}\). Therefore, \(\sin 7\pi/6 = -\frac{1}{2}\), and so \(\csc 7\pi/6 = \frac{1}{\sin 7\pi/6} = \frac{1}{-\frac{1}{2}}\).

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Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating any angle to an angle between 0 and 90 degrees (or 0 and π/2 radians). Using reference angles allows you to find exact trigonometric values without a calculator.

Trigonometric Functions and Their Signs in Different Quadrants

The sign of trigonometric functions depends on the quadrant in which the angle lies. For example, cosecant (csc) is positive in quadrants where sine is positive. Knowing the quadrant of the given angle helps determine the correct sign of the trigonometric value after using the reference angle.

Exact Values of Common Angles

Certain angles like π/6, π/4, and π/3 have well-known exact trigonometric values. For instance, sin(π/6) = 1/2, so csc(π/6) = 2. Recognizing these standard values is essential for finding exact answers without a calculator when working with reference angles.