Problem 58 - Algebra 2

Question

Which pairs of measurements represent the same angle measures? \(I.240^{\circ}, \frac{7 \pi}{6}\) radians \(\quad\) II. \(135^{\circ}, \frac{3 \pi}{4}\) radians \(\quad\) III. \(150^{\circ}, \frac{5 \pi}{6}\) radians A. I and II only \(\quad\) B. \(I\) and III only C. II and III only \(\quad\) D. I,II, and III

Step-by-Step Solution

Answer

The pairs II and III represent the same angle measures, hence the choice 'C' is correct.

Step 1: Convert degrees to radians

We will convert each angle from degrees to radians using the conversion factor \( \frac{\pi}{180} \). \nI. \( 240^{\circ} \)=\( 240 \times \frac{\pi}{180} \) = \( \frac{4 \pi}{3} \) radians \nII. \( 135^{\circ} \) = \( 135 \times \frac{\pi}{180} \) = \( \frac{3 \pi}{4} \) radians \nIII. \( 150^{\circ} \) = \( 150 \times \frac{\pi}{180} \) = \( \frac{5 \pi}{6} \) radians

Step 2: Compare the radians

Next we can compare the angles given in the question as radians to the ones we converted from degrees.\nI. \( \frac{7 \pi}{6} \) radians is not equal to \( \frac{4 \pi}{3} \) radians. \nII. \( \frac{3 \pi}{4} \) radians equals to \( \frac{3 \pi}{4} \) radians. \nIII. \( \frac{5 \pi}{6} \) radians equals to \( \frac{5 \pi}{6} \) radians.

Step 3: Identify the pairs

From the comparisons made in Step 2, we conclude that II and III pairs of measurements represent the same angle measures.

Key Concepts

Radians and Degrees

Degrees and radians are two different ways of measuring angles. The concept of degrees comes from dividing a circle into 360 parts, which is a very ancient and intuitive approach.

• One full circle is 360 degrees.
• Half a circle is 180 degrees.
• Quarter-circle is 90 degrees.

Radians, on the other hand, arise from considering the relationship between a circle and its radius. They give us a measure in terms of the radius instead of arbitrary degree units.

• One full circle is equal to \(2\pi\) radians.
• Half a circle is \(\pi\).
• A quarter-circle is \(\frac{\pi}{2}\).

Understanding both units is fundamental because they are used interchangeably in trigonometry, calculus, and various fields of engineering.

Angle Conversion

Converting between degrees and radians is crucial for solving trigonometric problems. The conversion factor to remember is \(\frac{\pi}{180}\). This factor helps in switching any angle given in degrees to radians.

• To convert from degrees to radians, multiply by \(\frac{\pi}{180}\).
• To convert from radians to degrees, multiply by \(\frac{180}{\pi}\).

Let's take an example: to convert \(240^{\circ}\) to radians, the calculation is \(240 \times \frac{\pi}{180} = \frac{4\pi}{3}\) radians.
Similarly, for \(135^{\circ}\), the conversion would be \(135 \times \frac{\pi}{180} = \frac{3\pi}{4}\). Practicing these conversions improves your ability to switch between units smoothly, making it easier to grasp problems involving angle measures.

Angle Measures

When working with angles, it's important to understand their measures, which can be given in either degrees or radians. Different contexts may require different units.

• Conventional protractors measure angles in degrees.
• Calculators, especially scientific ones, often require users to set the correct mode for calculations: degree or radian.

Some contexts, such as physics or engineering, favor radians because they provide a natural way to relate circular motion and angle measures to linear distances and speeds. In the given exercise, comparing angle measures involves converting degrees to radians because two angles are expressed in different units, ensuring that the form of measurement is consistent.
For instance, the angle \(135^{\circ}\) when converted correctly matches with its radian equivalent \(\frac{3\pi}{4}\), validating that the two express the same measurement. Understanding different angle measures and conversions allows seamless application across various problems in trigonometry.