Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\sin \frac{\theta}{2}=1$$

Step-by-Step Solution

Answer

\( \theta = \pi \) radians or \( \theta = 180 \) degrees.

Step 1: Understanding the Sine Equation

We start by understanding the equation \( \sin \left(\frac{\theta}{2}\right) = 1 \). The sine function reaches the value of 1 at an angle of \( \frac{\pi}{2} \) radians (or 90 degrees) within a period \( 0 \) to \( 2\pi \). We need \( \frac{\theta}{2} = \frac{\pi}{2} \) to satisfy \( \sin(\cdot) = 1 \).

Step 2: Solving for \(\theta\)

Next, solve the equation \( \frac{\theta}{2} = \frac{\pi}{2} \). By multiplying both sides by 2, we find \( \theta = \pi \). In degrees, this is equivalent to multiplying radians by \( \frac{180}{\pi} \), thus \( \theta = 180 \) degrees.

Step 3: Outputting the Solution

The solution to \( \sin\left( \frac{\theta}{2} \right) = 1 \) is found by solving for \( \theta \). Thus, \( \theta = \pi \) radians or \( \theta = 180 \) degrees. This angle measure is the nonnegative smallest angle for which the condition is satisfied.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, represented as \( \sin(\theta) \), where \( \theta \) is the angle. It defines the ratio of the length of the side opposite the angle over the hypotenuse in a right triangle. A cosine curve on the unit circle visually represents this function.
The sine function is periodic, meaning it repeats its values in regular intervals. Specifically, its period is \(2\pi\) radians or 360 degrees. It peaks at 1, which occurs at \( \frac{\pi}{2} \) radians (or 90 degrees) and bottoms out at -1 at \( \frac{3\pi}{2} \) radians (or 270 degrees).
Understanding the sine function's behavior over its period is crucial for solving trigonometric equations, like understanding where it equals specific values.

Radians and Degrees

Radians and degrees are both units to measure angles. Understanding how they interconvert is essential in trigonometry. Degrees divide a circle into 360 equal parts, making a complete circle 360 degrees. Whereas, radians measure angles by the length of the arc on the unit circle.
In radians, a complete circle is \(2\pi\) radians. Therefore, \( \pi \) radians equal 180 degrees. Conversion between these units can be done using the formula:

• From degrees to radians: Multiply by \( \frac{\pi}{180} \).
• From radians to degrees: Multiply by \( \frac{180}{\pi} \).

Mastering this conversion helps in solving trigonometric equations and interpreting results accurately.

Exact Solutions

Exact solutions in trigonometry are those where answers can be expressed in terms of fractions of \( \pi \) (for radians) or specific degrees without needing decimal approximations. These solutions are precise, unlike approximate solutions rounded to a certain decimal point.
For instance, the solution \( \theta = \pi \) radians or 180 degrees is an exact solution as these values are standard angles on the unit circle. Such solutions often arise when solving trigonometric equations where angles like \(0, \frac{\pi}{2}, \pi,\) etc., have significant interpretive meanings.
Differentiating between exact and approximate solutions ensures clear communication of the results and understanding of the problem conditions.

Angle Measures

Angle measures are the values that represent the rotation or opening between two lines at a common vertex. Understanding angle measures forms the basis of trigonometry and is essential for solving trigonometric problems.
Angles can be measured in:

• Degrees, ranging from 0 to 360, where 360 represents one complete rotation.
• Radians, where \(2\pi\) signifies a full circle.

Every trigonometric function, like the sine function, has outputs dependent on these angle measures. Forthe equation \( \sin\left(\frac{\theta}{2}\right) = 1 \), the angle analysis helps determine the smallest nonnegative angle measure where this equation holds.
By consistently using the smallest possible angle measures, solutions tend to be more easily understood and communicated.