Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cos \theta+1=0$$

Step-by-Step Solution

Answer

The solution is \( \theta = \pi \) radians or \( \theta = 180 \) degrees.

Step 1: Isolate Cosine

The given equation is \( \cos \theta + 1 = 0 \). To solve this, we first need to isolate the cosine function. Subtracting 1 from both sides, we have \( \cos \theta = -1 \).

Step 2: Solve for Theta in Radians

We need to find \( \theta \) values for which \( \cos \theta = -1 \). The standard nonnegative angle for which the cosine function equals \(-1\) is \( \theta = \pi \) radians. Since cosine is \(-1\) at \( \pi \) and not elsewhere in the domain \([0, 2\pi)\), the solution in radians is \( \theta = \pi \).

Step 3: Convert Radians to Degrees

To convert the radian measure \( \pi \) to degrees, we use the conversion formula: \( \theta_{degrees} = \theta_{radians} \times \frac{180}{\pi} \). Thus, the calculation becomes \( \pi \times \frac{180}{\pi} = 180 \) degrees.

Key Concepts

Radians to Degrees Conversion

Often, angles are expressed in radians or degrees, two units of measurement for angles. Converting between these units is a fundamental skill in trigonometry. The conversion formula from radians to degrees is:

• \( \theta_{degrees} = \theta_{radians} \times \frac{180}{\pi} \)

This is derived from the fact that \( 180 \) degrees is equivalent to \( \pi \) radians. Hence, multiplying radians by \( \frac{180}{\pi} \) converts it to degrees. For example, if we have an angle \( \pi \) radians, converting it to degrees using our formula gives:

• \( \pi \times \frac{180}{\pi} = 180 \) degrees

Converting radians to degrees allows for easier interpretation in everyday situations by using the more familiar degree measurement.

Cosine Function

The cosine function is a fundamental trigonometric function, closely linked to the unit circle definition. It measures the x-coordinate of a point formed by rotating a radius counterclockwise from the positive x-axis around the origin of a circle with radius 1. Key properties of the cosine function include:

• Its range: \([-1, 1]\)
• It is an even function, so \( \cos(-\theta) = \cos(\theta) \)
• The cosine function has a periodicity of \(2\pi\), repeating every full circle rotation

For the equation \( \cos \theta + 1 = 0 \), isolating \( \cos \theta \) gives \( \cos \theta = -1 \). This occurs at \( \theta = \pi \) in the principal range \([0, 2\pi)\). This angle points directly to the negative x-axis, a unique position where the cosine function achieves a value of \(-1\).

Nonnegative Angle Measure

In mathematics, especially in trigonometry, finding solutions within the smallest nonnegative angle measure is often desired. A nonnegative angle measure is simply an angle that is 0 or greater and often refers to angles measured in the counterclockwise direction from the positive x-axis. For periodic functions like cosine, one complete cycle is \([0, 2\pi)\) radians or \([0, 360)\) degrees. Solutions should be contained within this range whenever possible to ensure they are the smallest, simplest representations. In solving \( \cos \theta = -1 \), by focusing on the principal values in the range of \([0, 2\pi)\), we find the smallest nonnegative solution is at \( \theta = \pi \). Choosing nonnegative angle measures aids clarity and consistency, critical in solving trigonometric equations effectively.