Problem 7 - Contemporary Precalculus

Question

In Exercises \(1-10,\) use the definition (not a calculator) to find the function value. $$\cos (-3 \pi / 2)$$

Step-by-Step Solution

Answer

Answer: The value of the cosine function at the angle \(-3\pi/2\) is 0.

Step 1: Understand the definition of the cosine function

The cosine function is a periodic function that relates an angle to the x-coordinate of a point on the unit circle. The cosine function is defined as follows: For an angle \(\theta\) in standard position, the cosine of \(\theta\) is the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

Step 2: Identify the given angle

The given angle is \(-3\pi/2\). It represents a clockwise rotation by \(3\pi/2\) radians or \(270^\circ\).

Step 3: Locate the angle on the unit circle

Recall that the unit circle has a radius of 1 and is centered at the origin of the Cartesian coordinate plane. Start from the positive x-axis and rotate clockwise (negative rotation) by \(3\pi/2\), or \(270^\circ\). This rotation will take us to the positive y-axis.

Step 4: Find the x-coordinate of the intersection point

Since the terminal side of the \(-3\pi/2\) angle intersects the unit circle at the positive y-axis, the x-coordinate of this intersection point is 0.

Step 5: Determine the cosine function value

The cosine of the angle \(-3\pi/2\) is the x-coordinate of the intersection point, which is 0. So, \(\cos(-3\pi/2) = 0\).

Key Concepts

Unit Circle

The unit circle is an essential concept in trigonometry. It's a circle with a radius of 1 centered at the origin of the Cartesian coordinate plane. This simple structure allows us to explore relationships between angles and trigonometric functions like sine and cosine.

Here's why the unit circle is so crucial:

• Reference Tool: Every angle's sine and cosine can be visualized as coordinates (x, y) on this circle.
• Radian Measure: Angles are often measured in radians on the unit circle, providing a clear geometric interpretation.
• 360-Degree Coverage: The circle covers all possible rotations, making it perfect for studying full rotations seamlessly.

By relating the angle's position to a point on this circle, we derive trigonometric function values like the cosine. When you rotate along the circle, cosine values reflect the x-coordinates of these points.

Cosine Function

The cosine function connects angles with specific x-coordinates on the unit circle. It's a crucial trigonometric function, and understanding how it works is key to mastering trigonometry.

Key aspects of the cosine function include:

• Definition: Cosine of an angle \(\theta\) equals the x-coordinate of where the angle's terminal side intersects the unit circle.
• Periodic Nature: The cosine function is periodic with a cycle of \(2\pi\), meaning it repeats every \(360^\circ\).
• Range and Domain: The domain is all real numbers, and the range is \([-1, 1]\).

For the angle \(-3\pi/2\), we see this concept in action. By rotating clockwise \(270^\circ\) or \(3\pi/2\) radians on the unit circle, we find the x-coordinate of the intersection point with the circle is 0. Hence, \(\cos(-3\pi/2) = 0\).

Negative Angles

Negative angles can seem confusing at first, but they are simply rotations in the opposite direction, typically clockwise, on the unit circle. Here's how they fit into our understanding of trigonometry:

• Clockwise Rotation: While positive angles rotate counterclockwise, negative angles rotate clockwise.
• Reversibility: Negative and positive angles of the same magnitude fall at the same relative position but approach from opposite directions.
• Use of Reference Angles: Often, understanding the equivalent positive angle helps find the trigonometric value.

In our exercise, \(-3\pi/2\) is a negative angle, representing a \(270^\circ\) clockwise rotation. When visualized on the unit circle, it leads us to the positive y-axis where the x-coordinate—and hence the cosine—is 0.