Problem 69 - Algebra and Trigonometry

Question

use reference angles to find the exact value of each expression. Do not use a calculator. $$ \csc \frac{7 \pi}{6} $$

Step-by-Step Solution

Answer

The exact value of \( \csc \frac{7 \pi}{6} \) is -2.

Step 1: Determine the reference angle for \( \frac{7 \pi}{6} \)

We should first identify in which quadrant the angle \( \frac{7 \pi}{6} \) is located. Considering the unit circle, full circle equals to \(2 \pi\), so half of it is \(\pi\). This angle is more than \(\pi\) but less than \( \frac{3}{2} \pi \), meaning it falls in the 3rd quadrant. The reference angle for an angle in the 3rd quadrant can be determined by subtracting the angle from \(\pi\). In this case, it is \( \pi - \frac{7 \pi}{6} =-\frac{\pi}{6}\).

Step 2: Use the reference angle to calculate the sine value

In the 3rd quadrant, sine is negative. The sine of the reference angle \( \frac{\pi}{6} \) we know from the unit circle is \(\frac{1}{2}\). Therefore, the sine of \( \frac{7 \pi}{6} \) is -\(\frac{1}{2}\).

Step 3: Since cosecant is the reciprocal of sine, find the cosecant value

The cosecant is the reciprocal of the sine. So the cosecant of \( \frac{7 \pi}{6} \) would be the reciprocal of -\(\frac{1}{2}\), which is -2.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It helps in understanding angles and their corresponding coordinates on a circle with a radius of one. This circle is centered at the origin of the Cartesian plane. Remember, the full rotation of the unit circle is represented by \(2\pi\) radians.

In this exercise, we deal with the angle \( \frac{7\pi}{6} \). To determine its exact position and context, understanding the unit circle layout is crucial. When you place \( \frac{7\pi}{6} \) on the unit circle, it sits in the third quadrant. The unit circle helps indicate where each angle lands and what its reference angle is. Reference angles are typically measured from the x-axis.

For angles like \( \frac{7\pi}{6} \), knowing the unit circle for angles measured in radians allows for quick identification of their sine, cosine, and related functions.

Trigonometric Functions

Trigonometric functions are the building blocks for understanding angles and their properties. The most commonly used are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions relate the angles to the coordinates on the unit circle. For \(\frac{7\pi}{6}\), the sine and cosine functions are particularly useful for finding its cosecant (\(\csc\)).

The sine of an angle gives the y-coordinate on the unit circle, while the cosine gives the x-coordinate. In this scenario, knowing that the reference angle has a sine value of \(\frac{1}{2}\) simplifies calculations. Because it's in the third quadrant, the sine is negative, thus the value for \(\sin\left(\frac{7\pi}{6}\right)\) is \(-\frac{1}{2}\).

Reciprocal Identities

Reciprocal identities provide a way to find other trigonometric functions using basic ones like sine, cosine, and tangent. Each primary trigonometric function (except tangent and cotangent) has a reciprocal:

• Sine (\(\sin\)) has its reciprocal as cosecant (\(\csc\)).
• Cosine (\(\cos\)) reciprocates to secant (\(\sec\)).

The recipe for finding these reciprocals is straightforward. Take the sine value, which is \(-\frac{1}{2}\) for \(\frac{7\pi}{6}\), then flip it to find the cosecant. Hence, \(\csc\left(\frac{7\pi}{6}\right)\) is \(-2\).

This is a common method used to find exact values without complex calculations on a calculator.

Quadrants in the Cartesian Plane

The Cartesian plane helps divide the unit circle into four sections, known as quadrants. Each quadrant has a unique character based on the signs of coordinates of angles found there.

• First Quadrant: Both sine and cosine values are positive.
• Second Quadrant: Sine is positive, and cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Sine is negative, and cosine is positive.

The angle \(\frac{7\pi}{6}\) falls in the Third Quadrant, clarifying why its sine and cosine values are negative. This information is pivotal for evaluating trigonometric functions because each quadrant's properties dictate the sign and sometimes the value of these functions.