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Magazine Chapter 6: Problem 8
Find all solutions on the interval \(0 \leq \theta<2 \pi\). $$ \cos (\theta)=-1 $$
Short Answer
Expert verified
The solution is \(\theta = \pi\).
Step by step solution
01
Understanding the Problem
We need to find all values of \(\theta\) within the interval \(0 \leq \theta < 2\pi\) that satisfy the equation \(\cos(\theta) = -1\).
02
Recall the Unit Circle
On the unit circle, \(\cos(\theta)\) represents the x-coordinate of the point where the terminal side of angle \(\theta\) intersects the unit circle.
03
Locate where \(\cos(\theta) = -1\) on the Unit Circle
The cosine function reaches -1 at the angle where the point on the unit circle is on the negative x-axis. This occurs at the angle \(\theta = \pi\).
04
Verify the Condition in the Given Interval
Check that \(\theta = \pi\) is within the interval \([0, 2\pi)\). Since \(\pi\) is between 0 and \(2\pi\), it satisfies the interval condition.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
unit circle
The unit circle is a fundamental concept in trigonometry, featuring a circle with a radius of one unit centered at the origin of a coordinate plane. It is essential for understanding trigonometric functions such as sine, cosine, and tangent.
Here are some key features of the unit circle:
Here are some key features of the unit circle:
- Every point on the unit circle has coordinates \((\cos(\theta), \sin(\theta))\), where \(\theta\)\ is the angle formed by the positive x-axis and the line connecting the origin to the point.
- The angle \(\theta\)\ is measured in radians, a unit of angular measure convenient in math involving circles. Recall that \(2\pi\)\ radians corresponds to a full circle, equivalent to 360 degrees.
- On the unit circle, angles can be measured in both clockwise and counterclockwise directions, allowing for understanding of negative and positive angles.
- As \(\theta\)\ increases from 0 to \(2\pi\), the x-coordinate traces the cosine function, while the y-coordinate traces the sine function.
cosine function
The cosine function is one of the primary trigonometric functions, describing how the x-coordinate on the unit circle changes as an angle \(\theta\)\ varies. This function is key to solving many trigonometric problems.
- Cosine of an angle \(\theta\), denoted as \(\cos(\theta)\), represents the x-coordinate of the corresponding point on the unit circle.
- The range of the cosine function is from -1 to 1, as these are the boundaries along the x-axis for a circle of radius one.
- The cosine function is periodic and even, meaning it repeats every \(2\pi\)\ and is symmetric with respect to the y-axis. Thus, \(\cos(\theta) = \cos(-\theta)\).
- Key angles to remember include \(\theta = 0, \pi/2, \pi, 3\pi/2, \) and \(2\pi\), where \(\cos(\theta)\) respectively equals 1, 0, -1, 0, and 1.
solutions for theta
To find solutions for \(\theta\) in trigonometric equations like \(\cos(\theta) = -1\), we apply the properties of the cosine function and the unit circle.
- Identify the values of \(\theta\) from the unit circle where the cosine value matches the given condition.
- For the equation \(\cos(\theta) = -1\), refer to the unit circle where this condition is met. The point where the x-coordinate is -1 lies on the negative x-axis, which happens precisely at \(\theta = \pi\).
- Verify that \(\theta = \pi\) is within the given interval. For our example, \(0 \leq \theta < 2\pi\), \(\pi\) is a valid solution because it falls exactly within this range.
- Given the periodic nature of cosine, consider its periodicity of \(2\pi\) to check for other solutions within additional cycles, but here, we check only one full cycle.
