Problem 25 - Algebra 2 and Trigonometry

Question

In \(23-27,\) for each angle with the given radian measure: a. Give the measure of the angle in degrees. b. Give the measure of the reference angle in radians c. Draw the angle in standard position and its] reference angle as an acute angle formed by the terminal side of the angle and the \(x\) -axis. \(\frac{10 \pi}{9}\)

Step-by-Step Solution

Answer

a. 200 degrees; b. \(\frac{\pi}{9}\); c. Sketch in third quadrant with \(\frac{\pi}{9}\) as reference angle.

Step 1: Convert Radians to Degrees

First, convert the given radian measure to degrees. Use the formula \(\theta (\text{degrees}) = \theta (\text{radians}) \cdot \frac{180}{\pi}\). Substitute \(\frac{10\pi}{9}\) into the formula: \(\frac{10\pi}{9} \cdot \frac{180}{\pi} = \frac{10 \cdot 180}{9} = 200\) degrees.

Step 2: Determine the Reference Angle in Radians

The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. Since \(\frac{10\pi}{9}\) is greater than \(\pi\) but less than \(2\pi\), it lies in the third quadrant. Calculate the reference angle using \(\pi - \text{angle}\): \(\frac{10\pi}{9} - \pi = \frac{10\pi}{9} - \frac{9\pi}{9} = \frac{\pi}{9}\).

Step 3: Sketch the Angle and Reference Angle

1. Draw a coordinate plane with the x-axis and y-axis intersecting at the origin.2. Sketch the angle \(\frac{10\pi}{9}\) in standard position: starting from the positive x-axis, rotate counterclockwise past \(\pi\) (180 degrees, leftmost on the x-axis) to extend into the third quadrant (200 degrees).3. To draw the reference angle, identify the acute angle between the terminal side of \(\frac{10\pi}{9}\) and the negative x-axis. This angle is \(\frac{\pi}{9}\) in the third quadrant.

Key Concepts

Angle Conversion

Converting angles from radians to degrees is an essential skill in trigonometry and calculus. Angles are often measured in both degrees and radians, and knowing how to switch between these units can help in understanding various mathematical problems. The conversion formula is:
\[ \theta (\text{degrees}) = \theta (\text{radians}) \cdot \frac{180}{\pi} \]

• To convert from radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).
• For instance, converting \(\frac{10\pi}{9}\) radians to degrees involves calculating \(\frac{10\cdot 180}{9} = 200\) degrees.

Degrees describe how much of a circle an angle encompasses, where one complete circle is 360 degrees. Radians, on the other hand, measure the angle based on the circle's radius. Because there are \(2\pi\) radians in a circle, they offer a direct relationship with the arc lengths and are often preferred in calculus.

Reference Angles

A reference angle is the acute angle that a given angle forms with the x-axis. It is always positive and provides a simple way to understand angles that reach beyond the first quadrant.

• If an angle being measured is between 0 and \(\pi /2\), the reference angle is the angle itself.
• When an angle is between \(\pi /2\) and \(\pi\), subtract the angle from \(\pi\) to find the reference angle.
• In the case of \(\frac{10\pi}{9}\), which lies in the third quadrant, the reference angle is found by calculating \(\frac{10\pi}{9} - \pi = \frac{\pi}{9}\).

Reference angles are useful because they help to simplify complex calculations, especially when dealing with trigonometric functions. They ensure that the angle's trigonometric values in any quadrant can be related to those in the first quadrant.

Standard Position Angles

An angle is said to be in standard position when its vertex is at the origin of a coordinate plane, and its initial side lies along the positive x-axis. This standard form simplifies the evaluation and drawing of angles, making it a fundamental concept in trigonometry.

Steps to draw an angle in standard position:

• Place the angle's vertex at the origin (0,0).
• Ensure the initial side of the angle is along the positive x-axis.
• Rotate the terminal side counterclockwise to form the angle.

For \(\frac{10\pi}{9}\) radians, start from the positive x-axis and rotate into the third quadrant because the angle exceeds \(\pi\) radians but is less than \(2\pi\). The third quadrant is where both the sine and cosine of the angle are negative, so it's essential to note these characteristics when dealing with trigonometric functions.

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. It provides a natural mathematical approach to dealing with circular motion and trigonometric functions. Instead of dividing a circle into 360 parts, radian measure relates the angle to the circle's radius and circumference.

• There are \(2\pi\) radians in a complete circle, equating to 360 degrees.
• An angle of one radian occurs when the arc length equals the radius of the circle.
• This measurement is integral in calculus and physics, as it simplifies formulas and equations for circular motion.

An example is the angle \(\frac{10\pi}{9}\), which signifies that the arc length corresponding to the angle is \(\frac{10}{9}\) times longer than the circle's radius. This proportional method of measuring angles is why radians appear often in higher mathematics and should be understood thoroughly.