Problem 45 - Algebra 2

Question

Suppose \(\theta\) is an angle in standard position whose terminal side is in the given quadrant. For each function, find the exact values of the remaining five trigonometric functions of \(\theta .\) \(-\frac{2 \pi}{3}\)

Step-by-Step Solution

Answer

The remaining functions are \(\cos(\theta) = -\frac{1}{2}\), \(\tan(\theta) = \sqrt{3}\), \(\csc(\theta) = -\frac{2\sqrt{3}}{3}\), \(\sec(\theta) = -2\), and \(\cot(\theta) = \frac{\sqrt{3}}{3}\).

Step 1: Identify the Given Angle

The angle is given as \(-\frac{2\pi}{3}\). This angle is in standard position and we need to locate it within a unit circle.

Step 2: Determine the Angle's Quadrant

The angle \(-\frac{2\pi}{3}\) converts to \(-120^\circ\). In a unit circle, negative angles rotate clockwise. Since \(-120^\circ\) places us 120 degrees clockwise from the positive x-axis, this lies in Quadrant III.

Step 3: Identify Reference Angle

The reference angle \(\theta_{ref}\) is the acute angle made by the terminal side of \(\theta\) with the horizontal axis. For \(-120^\circ\), the reference angle is \(180^\circ - 120^\circ = 60^\circ\) or \( \frac{\pi}{3} \) radians.

Step 4: Evaluate Sine and Cosine Functions

In Quadrant III, the sine and cosine functions are both negative. Using the reference angle, we know \(\sin(\theta) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\) and \(\cos(\theta) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2}\).

Step 5: Determine Other Trigonometric Functions

With \(\sin(\theta) = -\frac{\sqrt{3}}{2}\) and \(\cos(\theta) = -\frac{1}{2}\), we calculate the other functions:- \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}\)- \(\csc(\theta) = \frac{1}{\sin(\theta)} = -\frac{2}{\sqrt{3}}\), which simplifies to \( -\frac{2\sqrt{3}}{3} \)- \(\sec(\theta) = \frac{1}{\cos(\theta)} = -2\)- \(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{\sqrt{3}}\), which simplifies to \( \frac{\sqrt{3}}{3} \).

Key Concepts

Angle in Standard Position

An angle in standard position is one that has its vertex at the origin of a coordinate plane. This means the initial side of the angle lies along the positive x-axis. When measuring the angle, if it is positive, it moves counter-clockwise from the x-axis. Conversely, a negative angle, like \(-\frac{2\pi}{3}\), moves clockwise. This concept simplifies the visualization of angles and their interaction with trigonometric functions as their position is standardized based on the x-axis.

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle with the radius of one, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to an angle from the x-axis starting from 0 degrees or 0 radians. The coordinates of each point can help compute the sine and cosine values corresponding to those angles.

• Radius = 1
• Center = (0,0)

Using the unit circle, negative angles, such as \(-\frac{2\pi}{3}\), can be rotated clockwise, making quadrant identification straightforward.

Reference Angle

The reference angle is the smallest angle that the terminal side of the given angle makes with the x-axis. It is always positive and less than or equal to 90 degrees, or \(\frac{\pi}{2}\) radians. The reference angle is found by subtracting the angle from the nearest multiple of 180 degrees if necessary. For example, the reference angle here for \(-120^\circ\) is found by calculating \(180^\circ - 120^\circ = 60^\circ\). This helps in finding the sine and cosine of an angle.

Quadrant Identification

Quadrants are sections of the coordinate plane. Starting from the positive x-axis moving counterclockwise, they are numbered I through IV. Each quadrant dictates the sign of the trigonometric functions:

• Quadrant I: All are positive
• Quadrant II: Sine is positive, cosine is negative
• Quadrant III: Sine and cosine are negative
• Quadrant IV: Cosine is positive, sine is negative

The angle \(-\frac{2\pi}{3}\), when translated to \(-120^\circ\), falls in Quadrant III. This means both sine and cosine values here will be negative.

Sine and Cosine Values

Sine and cosine are primary trigonometric functions critical to understanding angle measures. These functions are directly related to the unit circle:

• Sine of angle \(\theta\) gives the y-coordinate
• Cosine of the angle gives the x-coordinate

For a reference angle of \( \frac{\pi}{3}\), from the unit circle, \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\) and \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\). In Quadrant III, both of these will be negative:

• \(\sin(\theta) = -\frac{\sqrt{3}}{2}\)
• \(\cos(\theta) = -\frac{1}{2}\)

These values are key to finding the remaining trigonometric functions.