Problem 41 - Precalculus

Question

Use the unit circle to find all of the exact values of \(\theta\) that make the equation true in the indicated interval. $$\cos \theta=-1,0 \leq \theta \leq 4 \pi$$

Step-by-Step Solution

Answer

\(\theta = \pi\) and \(\theta = 3\pi\).

Step 1: Understanding the Cosine Function

The cosine function \, \(\cos \theta\), is the horizontal coordinate of points on the unit circle. For \(\cos \theta = -1\), \(\theta\) is at a point where the circle intersects the negative side of the x-axis.

Step 2: Identifying Initial Solutions

In the unit circle, \(\cos \theta = -1\) at \(\theta = \pi\) because this is where the point is (-1, 0) on the unit circle.

Step 3: Finding Additional Solutions in the Interval

Since the interval is from \(0\) to \(4\pi\), and the cosine function repeats every \(2\pi\), we check increments by \(2\pi\) to find additional solutions. For \(\theta = 3\pi\), \(\cos(3\pi) = -1\).

Step 4: Verifying the Interval

Verify that all solutions \(\pi\) and \(3\pi\) lie within the interval \([0, 4\pi]\). The values of \(\theta\) are within the given range, covering the first full cycle and halfway into the second.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos \theta \), plays a crucial role in trigonometry, especially when dealing with the unit circle. It helps in understanding the relationship between angles and points on a circle with a radius of one unit. In simple terms, the cosine of an angle \( \theta \) is the horizontal distance from the origin to a point on the circle corresponding to that angle.
Understanding the cosine function:

• The cosine value ranges from -1 to 1.
• \( \cos(0) = 1 \), meaning at an angle of 0, the distance is maximum,(equal to the radius of 1)
• \( \cos(\pi) = -1 \), indicating the extreme left on the x-axis.

When \( \cos \theta = -1 \), it means our point on the unit circle is at (-1, 0), exactly opposite to the starting point. Importantly, the cosine function is periodic, repeating every \( 2\pi \) radians, making it crucial to identify solutions over given intervals by considering this periodic nature.

Angle Intervals

In trigonometry, angles can be measured in degrees or radians, and they can span different intervals. When working with trigonometry problems, it's common to see angles being restricted within certain bounds, which are called angle intervals.
An interval specifies the range of angles for which solutions are to be found.

• An interval like \( 0 \leq \theta \leq 4\pi \) defines a starting angle from 0 up through \( 4\pi \) radians.
• This interval represents two full circles (as \( 2\pi \) makes one full circle) plus another half circle.

When searching for solutions within a specific interval, such as \( 0 \leq \theta \leq 4\pi \), it’s important to take into account the periodic nature of trigonometric functions. The cosine value will repeat every \( 2\pi \), so solutions must be checked by adding or subtracting \( 2\pi \) intervals to find all valid angles that satisfy the given condition within the specified range.

Exact Values of Angles

When discussing trigonometric functions, exact values of angles become significant. These are specific angles for which the trigonometric values (like sine, cosine, and tangent) are known precisely.

• For instance, \( \pi \) and \( 3\pi \) are key angles for which \( \cos(\pi) = -1 \) and \( \cos(3\pi) = -1 \).
• These values occur where the unit circle's point is exactly on the negative x-axis.

The unit circle approach involves looking at circle intersections that provide these clear and direct values.Finding exact angle values requires understanding both the unit circle and how angles repeat:

• On the unit circle, specific angles such as \( \pi \) and \( 3\pi \) are derived directly from the symmetry and geometry of the circle.
• Knowing these distinct angles helps in solving trigonometric equations accurately and efficiently.

Working within intervals like \( 0 \leq \theta \leq 4\pi \), one identifies every solution accurately by addressing these exact points, ensuring a complete and proper solution.