Problem 56 - Contemporary Precalculus

Question

Use a unit circle diagram to explain why the given statement is true. $$\lim _{t \rightarrow \pi / 2} \cos t=0$$

Step-by-Step Solution

Answer

Question: Explain why the statement $\lim _{t \rightarrow \pi / 2} \cos t=0$ is true using a unit circle diagram. Answer: When we approach the angle π/2 in the unit circle, the x-coordinate (cosine value) of the point where the terminal side intersects the circle gets closer and closer to 0. This is because at π/2, the angle intersects the unit circle at the point (0,1), where the x-coordinate is 0. As the angle approaches π/2, the cosine value approaches 0, which is represented by the limit statement $\lim _{t \rightarrow \pi / 2} \cos t=0$.

Step 1: Understand the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a Cartesian coordinate system. The cosine of an angle θ, in relation to the unit circle, represents the x-coordinate of the point where the terminal side of angle θ intersects the circle. The sine of an angle θ represents the y-coordinate of the same point.

Step 2: Draw the Unit Circle and mark π/2

Draw the unit circle on the coordinate plane. Mark the angle π/2, or 90 degrees, in the first quadrant. This angle is formed when we move a radius along the circle from the positive x-axis counterclockwise until it is perpendicular to the x-axis. The terminal side of the angle π/2 will be lying on the positive y-axis.

Step 3: Identify the point where the terminal side of π/2 intersects the unit circle

When the angle is π/2, the terminal side of the angle will intersect the unit circle at the point (0,1). This is the point where the y-coordinate is at its maximum value of 1 but the x-coordinate is equal to 0.

Step 4: Determine the cosine value at π/2

As mentioned before, the cosine of an angle in the unit circle is the x-coordinate of the point where the terminal side intersects the circle. So, in this case, where the angle is π/2 and intersects the unit circle at (0,1), the cosine value is 0.

Step 5: Explain the limit statement using the unit circle diagram

As we approach the angle π/2 in the unit circle, the x-coordinate (cosine value) of the point where the terminal side intersects the circle gets closer and closer to 0. Therefore, the limit statement is true: when the angle approaches π/2, the cosine value approaches 0, which can be written as: $$\lim _{t \rightarrow \pi / 2} \cos t=0$$

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry and is closely associated with the unit circle. It helps us understand the horizontal component of an angle when represented on a circle with a radius of 1.

Here's how it works:

• On the unit circle, each angle has a corresponding point on the circumference.
• The x-coordinate of this point gives us the cosine of that angle.
• For example, at an angle of 0 degrees (or 0 radians), the point is at (1, 0) because the line stays on the positive x-axis. So, the cosine is 1.
• As the angle increases, the point moves around the circle. The x-coordinate changes, reflecting how cosine values evolve with the angle.

In practical terms, the cosine function helps determine values related to angles in various applications like physics, computer graphics, and even engineering. By understanding this function through the unit circle, we can visualize and comprehend these changing values.

Limits

The concept of limits provides a way to understand behavior in calculus, particularly how functions behave as they approach a particular point. In the context of the cosine function and the unit circle, the limit helps us see what happens to the cosine value as we get closer to a specific angle like \( \pi / 2 \).

Key aspects of limits include:

• They describe how a function behaves as an input nears a certain value, rather than getting the function’s value at that point.
• In our case, we're examining \( \lim _{t \rightarrow \pi / 2} \cos t \), showcasing how the cosine value approaches 0 as the angle approaches 90 degrees.
• This limit tells us about the behavior just before reaching \( \pi /2 \), even though the cosine of \( \pi / 2 \) itself is exactly 0, reflecting the x-coordinate of point (0,1) on the unit circle.

Limits are crucial for understanding continuous change and are foundational in calculus, helping solve real-world problems involving rates of change and motion.

Trigonometry

Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles. When applied to the unit circle, trigonometry helps us understand how angle measures influence both sine and cosine values.

Trigonometry offers several essential insights:

• It applies the principles of angles and circles not only to geometry but to real-world phenomena like sound waves, light, and more.
• In the unit circle, trigonometry shows how the sine and cosine values are derived from the coordinates of points on the circle's circumference.
• This is particularly evident for angles like \( \pi / 2 \), where the terminal side’s intersection with the unit circle results in coordinates (0,1), providing cosine as 0 and sine as 1.

By mastering trigonometry using the unit circle, students gain a practical understanding of how angles relate to circles, informing a wide range of scientific and engineering fields. Whether they are analyzing waves or measuring distances, the consistent application of these trigonometric principles offers valuable insights.