Problem 10 - Precalculus with Calculus Previews

Question

In Problems \(1-16\), draw the given angle in standard position. Bear in mind that the lack of a degree symbol \(\left(^{\circ}\right)\) in an angular measurement indicates that the angle is measured in radians. $$ 5 \pi $$

Step-by-Step Solution

Answer

The angle \(5\pi\) ends at the position of \(\pi\) radians in standard position.

Step 1: Understand the Angle

The given angle is \(5\pi\) radians. Note that 1 complete revolution around a circle is equal to \(2\pi\) radians.

Step 2: Calculate Full Circle Revolutions

To identify how many full circle revolutions the angle \(5\pi\) makes, divide \(5\pi\) by \(2\pi\): \[\frac{5\pi}{2\pi} = \frac{5}{2} = 2.5\]This result means that \(5\pi\) completes 2.5 full revolutions.

Step 3: Determine Equivalent Angle

Since \(5\pi\) completes 2 full revolutions (equivalent to \(4\pi\)), subtract \(4\pi\) from \(5\pi\) to find the equivalent angle within the first full revolution:\[5\pi - 4\pi = \pi\]

Key Concepts

Radians

Radians are a unit of measuring angles, just like degrees. However, they are based on the distance traveled around the circumference of a circle. This fundamental unit directly relates to the concept of pi (\(\pi\)), which is the ratio of a circle's circumference to its diameter. There are 2\(\pi\) radians in a full circle.

• One radian is the angle created when the radius is wrapped along the edge of the circle.
• The entire circle, therefore, measures 2\(\pi\) radians because the circle's circumference is \(2\pi\) times the radius.

When dealing with radians, it’s important to note that radians do not typically have a "degree" symbol. This is why the absence of this symbol in a problem often indicates that the angle is provided in radians.

Angle measurement

Angles can be measured in degrees or radians, offering different perspectives on the same angle. Degrees divide a circle into 360 equal parts, whereas radians use the radius of the circle as a measurement unit. Knowing how to convert between the two is crucial for solving physics and mathematics problems.

• To convert from radians to degrees, use the formula: \(\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\).
• To convert from degrees to radians, the formula is: \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\).

Remember that a straight angle is \(\pi\) radians or 180 degrees. Understanding these conversions will help in visualizing angles in both unit systems, making it easier to switch as necessary for various problems.

Standard position of angles

The standard position of an angle in trigonometry is when its vertex is positioned at the origin of a coordinate plane, on the point (0,0), and the initial side of the angle lies along the positive x-axis.

• From this position, positive angles are measured counter-clockwise.
• Negative angles are measured clockwise from the positive x-axis.

For example, the angle \(5\pi\) refers to full revolutions plus an angle. In this case, \(5\pi\) radians revolve around the circle 2.5 times (full revolutions), equating to an equivalent position as a \(\pi\) radian angle. Knowing the standard position assists in plotting and understanding angles on the coordinate plane.