Problem 48 - Precalculus with Calculus Previews

Question

Find all angles \(t,\) where \(0 \leq t<\) \(2 \pi,\) that satisfy the given condition. $$ \cos t=-1 $$

Step-by-Step Solution

Answer

The angle that satisfies \( \cos t = -1 \) is \( t = \pi \).

Step 1: Understanding the Problem

We need to find the values of angle \( t \) within the interval \( [0, 2\pi) \) such that \( \cos t = -1 \). This means we need to determine when the cosine of the angle equals -1.

Step 2: Analyzing the Unit Circle

On the unit circle, the cosine of an angle corresponds to the x-coordinate. Thus, \( \cos t = -1 \) means we need to find the angle where the x-coordinate is -1.

Step 3: Identifying the Angle

The angle where the x-coordinate is -1 on the unit circle is precisely at the angle \( t = \pi \). This is because at \( t = \pi \), we are directly to the left on the negative x-axis.

Step 4: Confirming the Solution Within Given Interval

We need to confirm that \( t = \pi \) falls within the interval \( [0, 2\pi) \), which it does. Hence, it is a valid solution.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of one, centered at the origin of a coordinate plane. This circle is significant because it provides a visual framework for understanding trigonometric functions. The coordinates of any point on the unit circle are represented as \(x, y\), where \ x \ is the cosine of the angle, and \ y \ is the sine of the angle.

On the unit circle, an angle \ t \ is measured from the positive x-axis, counterclockwise for positive angles and clockwise for negative angles. The circumference of the unit circle is naturally \(2\pi\), which covers one full rotation. Angles can be positive or negative depending on their direction around the circle. When dealing with the unit circle, every angle corresponds to a specific point with a unique coordinate on the circle.

• Unit circle helps in simplifying calculations in trigonometry.
• It's used to define the sine, cosine, and tangent functions for all real numbers.
• Each point on the circle corresponds to an angle and its cosine and sine values.

This powerful tool allows us to solve trigonometric equations and understand angles deeply.

Cosine Function

The cosine function, denoted as \ \cos(t) \, is one of the primary trigonometric functions. It is derived from the unit circle, specifically relating to the x-coordinate of points along the circle.

This function is periodic, meaning it repeats values at regular intervals. The basic period of the cosine function is \(2\pi\). Within this cycle, the cosine function starts at 1, decreases to -1, and returns to 1, forming a smooth wave.

• At \(t = 0\), \ \cos(0) = 1 \.
• At \(t = \pi/2\), \ \cos(\pi/2) = 0 \.
• At \(t = \pi\), \ \cos(\pi) = -1 \.

Understanding this behavior is crucial for solving trigonometric equations. In our problem, \ \cos(t) = -1 \ implies we are directly at \(t = \pi\), the half-circle mark on the negative x-axis.

Angle Measurement

Angle measurement is essential in both geometry and trigonometry. There are two common units for measuring angles: degrees and radians. Radians are often the preferred unit in higher mathematics, particularly in trigonometry, because they provide a natural way to describe angles based on the unit circle.

One complete rotation around the unit circle corresponds to \ 2\pi \ radians or 360 degrees. This means that \ \pi \ radians is equivalent to 180 degrees. Thus, when measuring angles in the unit circle:

• \(0\) radians corresponds to \(0\) degrees.
• \(\pi/2\) radians corresponds to \(90\) degrees.
• \(\pi\) radians corresponds to \(180\) degrees.

It's important to understand these conversions, as they allow you to switch between radians and degrees depending on the context of the problem. This understanding ensures that you correctly apply and interpret trigonometric functions, like the cosine in relation to angles, as seen in our exercise where \( t = \pi \) solves \ \cos t = -1 \.