Question

Given an angle \(\theta\) in standard position with its terminal side in Quadrant III, explain how you can use a reference angle to find \(\cos \theta\).

Short answer

To find the cosine of an angle located in the third quadrant, use the reference angle. However, keep in mind that the cosine value will be negative as both sine and cosine are negative in the third quadrant.

Step-by-step solution

Understanding Quadrants

Recognize that an angle in standard position is positioned at the origin with its initial side along the positive x-axis. The terminal side is in Quadrant III, meaning it's below the x-axis and to the left of the y-axis. In this quadrant, both \(\cos \theta\) and \(\sin \theta\) are negative.

Define the Reference Angle

The reference angle is always positive and is formed by the terminal side of the angle \(\theta\) and the x-axis. It's the acute angle within the right triangle formed. Let's denote the reference angle as \(\theta '\).

Find cosine using Reference Angle

Since the reference angle and the given angle have the same cosine value in magnitude but differ in sign based on the quadrant, the cosine of the angle \(\theta\) in Quadrant III is the same as the negative cosine of the reference angle \(\theta '\): \(\cos \theta = -\cos \theta '\).

Key concepts

Cosine in Different Quadrants

When learning trigonometry, understanding how the cosine function behaves in different quadrants is a key tool for solving problems. In the coordinate plane, the sign of the cosine function depends on the quadrant in which an angle in standard position terminates.

In Quadrant I, both the x-coordinate and the cosine values are positive. As we rotate counter-clockwise, angles that terminate in Quadrant II will have a negative cosine, since the x-coordinate is now negative. In Quadrant III, where both x and y coordinates are negative, the cosine values continue to be negative. Finally, in Quadrant IV, despite the y-coordinate being negative, the x-coordinate—and consequently the cosine—returns to being positive.

Remembering the signs of trigonometric functions in each quadrant can be simplified using the mnemonic 'All Students Take Calculus', which stands for All (positive in Quadrant I), Sin (positive in Quadrant II), Tangent (positive in Quadrant III), Cosine (positive in Quadrant IV). This helps to determine the sign of cosine when dealing with angles in various quadrants.

Standard Position of an Angle

To make sense of angles and their trigonometric functions, it's essential to place them in what is known as the standard position. An angle is said to be in standard position when its vertex is at the origin of the coordinate system and its initial side lies along the positive x-axis. From there, the angle opens in a counter-clockwise direction.

The standard position allows us to more easily classify angles based on where their terminal side falls and use this as a reference to understand trigonometric function values associated with those angles. For example, if the terminal side of an angle is in Quadrant III, we already know that its sine and cosine values will be negative, which is directly related to the position of the terminal side relative to the x and y axes.

Quadrant III Characteristics

Quadrant III is unique due to its specific characteristics that influence trigonometric functions. In this quadrant, both the x-coordinate and y-coordinate of any point are negative because the terminal side of an angle lies below the x-axis and to the left of the y-axis.

Consequently, for angles with their terminal side in Quadrant III, the cosine and sine values are both negative. This is crucial to understand because it impacts the calculation of trigonometric functions. When dealing with an angle in Quadrant III, remember that you are working in an area where both horizontal and vertical components are in the negative direction, affecting the signs of your results.

Using Reference Angles to Find Trigonometric Functions

A reference angle is an acute angle that a terminal side makes with the x-axis. It is always positive and can provide a straightforward way to find trigonometric functions for angles in any quadrant.

In the context of the Quadrant III example from our exercise, to find \(\cos \theta\), one would first determine the reference angle \(\theta ' \) that the terminal side of \(\theta\) forms with the x-axis. The cosine of \(\theta\) is equal to the negative cosine of \(\theta ' \) due to the terminal side of \(\theta\) being in Quadrant III: \(\cos \theta = -\cos \theta '\).

This approach simplifies calculations when working with angles beyond the first quadrant—understanding the concept of reference angles allows for quick and accurate computations of trigonometric values for these angles.