Problem 41 - Algebra 2 and Trigonometry

Question

Complete the following table, expressing degree measures in radian measure in terms of \(\pi\) $$ \begin{array}{|c|c|c|c|c|c|c|}\hline \text { Degrees } 30^{\circ} & {45^{\circ}} & {60^{\circ}} & {90^{\circ}} & {180^{\circ}} & {270^{\circ}} & {360^{\circ}} \\ \hline \text { Radians } & {} & {} & {} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

30° is \(\frac{\pi}{6}\), 45° is \(\frac{\pi}{4}\), 60° is \(\frac{\pi}{3}\), 90° is \(\frac{\pi}{2}\), 180° is \(\pi\), 270° is \(\frac{3\pi}{2}\), and 360° is \(2\pi\).

Step 1: Understand the Conversion

To convert degrees to radians, use the formula: \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \] This formula relates the degree measure to its equivalent in radians.

Step 2: Convert 30 Degrees to Radians

Apply the conversion formula to 30°:\[ 30^{\circ} \times \frac{\pi}{180} = \frac{30\pi}{180} = \frac{\pi}{6} \]Hence, 30° is equivalent to \( \frac{\pi}{6} \) in radians.

Step 3: Convert 45 Degrees to Radians

Apply the conversion formula to 45°:\[ 45^{\circ} \times \frac{\pi}{180} = \frac{45\pi}{180} = \frac{\pi}{4} \]So, 45° is equivalent to \( \frac{\pi}{4} \) in radians.

Step 4: Convert 60 Degrees to Radians

Convert 60° using the formula:\[ 60^{\circ} \times \frac{\pi}{180} = \frac{60\pi}{180} = \frac{\pi}{3} \]Thus, 60° is \( \frac{\pi}{3} \) in radians.

Step 5: Convert 90 Degrees to Radians

For 90°:\[ 90^{\circ} \times \frac{\pi}{180} = \frac{90\pi}{180} = \frac{\pi}{2} \]Therefore, 90° is \( \frac{\pi}{2} \) in radians.

Step 6: Convert 180 Degrees to Radians

Converting 180°:\[ 180^{\circ} \times \frac{\pi}{180} = \pi \]So, 180° is \( \pi \) in radians.

Step 7: Convert 270 Degrees to Radians

For 270°:\[ 270^{\circ} \times \frac{\pi}{180} = \frac{270\pi}{180} = \frac{3\pi}{2} \]Thus, 270° is \( \frac{3\pi}{2} \) in radians.

Step 8: Convert 360 Degrees to Radians

Finally, 360°:\[ 360^{\circ} \times \frac{\pi}{180} = 2\pi \]Therefore, 360° corresponds to \( 2\pi \) in radians.

Key Concepts

Degree to Radian Conversion

Understanding how to convert degrees into radians is a fundamental skill in trigonometry. Degrees and radians are two different units for measuring angles, much like miles and kilometers for distance.

To convert an angle from degrees to radians, we use the formula:
\[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \] This formula works because there are \(180^{\circ}\) in \(\pi\) radians, which is the same thing as saying \(360^{\circ}\) is equal to \(2\pi\) radians—like saying a full circle can be measured in both units.

When you multiply the degree measure by \(\frac{\pi}{180}\), it effectively scales down the measurement from a scale based on 360 to one based on \(2\pi\), thus converting degrees to radians. This means that for every degree, we have \(\frac{\pi}{180}\) radians.​ Some common conversions include:

• \(30^{\circ} = \frac{\pi}{6}\)
• \(45^{\circ} = \frac{\pi}{4}\)
• \(90^{\circ} = \frac{\pi}{2}\)
• \(180^{\circ} = \pi\)
• \(360^{\circ} = 2\pi\)

With practice, these conversions become almost second nature and can greatly aid problem-solving in trigonometry.

Trigonometric Angles

Trigonometry often involves working with special angles: \(30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ},\) and their respective multiples. These angles appear quite frequently in various mathematical problems, especially when dealing with triangles and circular functions.

Here is why they are considered 'special':

• They often correspond to simple fractions of \(\pi\) when converted to radians.
• They appear in real-world applications like navigation and physics.

When looking at a unit circle—a circle with a radius of 1—these angles map out key points:

• \(0^{\circ}\) or \(360^{\circ}\) maps to the positive x-axis.
• \(90^{\circ}\) maps to the positive y-axis.
• \(180^{\circ}\) maps to the negative x-axis.
• \(270^{\circ}\) maps to the negative y-axis.

Each of these angles reflects specific properties of sine, cosine, and tangent functions, which are fundamental to solving many trigonometric problems. Knowing these properties and their equivalent radian measures can simplify solving equations and understanding how triangles and circles behave.

Mathematical Formulas

Mathematical formulas are the building blocks that allow us to handle a variety of problems in trigonometry and beyond. Being comfortable with using them helps to break down and solve complex issues systematically.

In the context of converting degrees to radians, understanding and applying the correct formula is crucial. Here's a quick recap of how the formula works:
1. **Multiply the degree measure by the factor** \(\frac{\pi}{180}\): This is the fundamental operation that translates an angle from its degree measure to its radian counterpart.
2. **Simplify the expression**: Once multiplied, you'll get a fractional value, which may possibly need simplification to show the relationship in a form involving \(\pi\) and integers.
Here’s why formulas really matter:

• They provide a universal language for describing mathematical ideas.
• They're consistent, meaning the same approach works every time, minimizing errors.

Being adept with these conversion formulas equips students with the ability to move seamlessly between degrees and radians, a skill that is valuable not only in mathematics but also in fields like engineering and science.