Problem 78 - Precalculus: Functions and Graphs

Question

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and the terminal side of \(\boldsymbol{\theta}\) is in the specified quadrant and satisfles the given condition. III; bisects the quadrant

Step-by-Step Solution

Answer

The six trigonometric functions for \(\theta = 225\) degrees are \( \sin(\theta) = -\frac{\sqrt{2}}{2} \), \( \cos(\theta) = -\frac{\sqrt{2}}{2} \), \( \tan(\theta) = 1 \), \( \cot(\theta) = 1 \), \( \sec(\theta) = -\sqrt{2} \), \( \csc(\theta) = -\sqrt{2} \).

Step 1: Understanding the Quadrant and Angle Position

The problem states that the angle \( \theta \) is in Quadrant III. It's also given that \( \theta \) bisects the quadrant. Since a right angle is 90 degrees or \( \frac{\pi}{2} \) radians, each quadrant represents 90 degrees. Therefore, the angle bisecting Quadrant III occurs at an angle of 225 degrees or \( \frac{5\pi}{4} \) radians.

Step 2: Identify Reference Angle

To find the reference angle in Quadrant III for an angle of 225 degrees, subtract 180 degrees from 225 degrees. The reference angle \( \theta_{ref} \) is 45 degrees or \( \frac{\pi}{4} \) radians, which corresponds to the common reference angle known in the unit circle.

Step 3: Determine the Sine and Cosine

In Quadrant III, both sine and cosine values are negative. For the reference angle \( \frac{\pi}{4} \), the sine and cosine values are both \( \frac{\sqrt{2}}{2} \) in Quadrant I. Therefore, in Quadrant III, \( \sin(\theta) = -\frac{\sqrt{2}}{2} \) and \( \cos(\theta) = -\frac{\sqrt{2}}{2} \).

Step 4: Calculate the Tangent and Cotangent

Tangent is defined as the ratio of sine over cosine. Thus, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 \). Cotangent is the reciprocal of tangent, so \( \cot(\theta) = 1 \).

Step 5: Find the Secant and Cosecant

Secant is the reciprocal of cosine, so \( \sec(\theta) = -\frac{2}{\sqrt{2}} = -\sqrt{2} \). Cosecant is the reciprocal of sine, thus \( \csc(\theta) = -\sqrt{2} \).

Key Concepts

Quadrant III

In trigonometry, the coordinate plane is divided into four quadrants, each spanning 90 degrees. When we talk about Quadrant III, we're referring to the section of the coordinate plane where both x and y coordinates are negative. This quadrant covers the range from 180 to 270 degrees or from \( \pi \) to \( \frac{3\pi}{2} \) radians.

Placing an angle in Quadrant III means that the angle's terminal side has moved clockwise past 180 degrees but not beyond 270 degrees.

Some characteristics of angles in Quadrant III include:

• Both sine and cosine are negative.
• Tangent and cotangent, which are ratios involving sine and cosine, are positive due to the negative divided by negative rule.

Understanding where an angle lies is the first step in determining the values of its trigonometric functions.

reference angle

The reference angle is critical in simplifying trigonometric calculations. It's the smallest angle formed by the terminal side of the given angle and the x-axis, always taken between 0 and 90 degrees.

To find the reference angle for any given angle \( \theta \) that lies in Quadrant III, you subtract 180 degrees from the original angle. This is because the reference angle reflects how far the terminal side is from the x-axis.

In the exercise's case, with an angle of 225 degrees, the reference angle is \( \theta_{ref} \) = 225 - 180 = 45 degrees. In radians, this is equivalent to \( \frac{\pi}{4} \). This calculation is useful as it often leads to values that appear frequently in trigonometric tables, making further calculations easier.

sine and cosine

Sine and cosine are fundamental trigonometric functions associated with right-angled triangles. In the unit circle context, sine represents the y-coordinate, and cosine represents the x-coordinate of a point on the circle corresponding to an angle.

In Quadrant III, both sine and cosine values are negative. For example, with the reference angle of \(\pi/4\) in Quadrant III:

• The sine value is \( -\sqrt{2}/2 \).
• The cosine value is \( -\sqrt{2}/2 \).

This is due to the symmetry and the properties of trigonometric functions in different quadrants.

Understanding these negative signs is crucial, as they will affect other calculations like tangents, cotangents, and reciprocals such as secants and cosecants.

unit circle

The unit circle is a circle with a radius of 1 centered at the origin (0,0) of the coordinate plane. It is an essential tool in trigonometry because it relates angle measures in radians to coordinates and trigonometric function values.

On the unit circle, angles are measured from the positive x-axis with counterclockwise movement being positive.

• The x-coordinate corresponds to the cosine of the angle.
• The y-coordinate corresponds to the sine of the angle.

For angles in Quadrant III, both coordinates are negative, which explains why both sine and cosine values are negative for angles in this quadrant.

Knowing how to place an angle on the unit circle helps understand and remember the values for trigonometric functions and their behaviors, providing a powerful visual aid for solving trigonometric problems.