Problem 24 - 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. \(\frac{7 \pi}{36}\)

Step-by-Step Solution

Answer

Angle in degrees is 35°, and the reference angle is \( \frac{7\pi}{36} \) radians.

Step 1: Convert Radians to Degrees

To convert an angle from radians to degrees, use the formula \( \,\theta_{degrees} = \theta_{radians} \times \frac{180}{\pi} \, \). For \( \frac{7\pi}{36} \,\), this becomes: \[ \theta_{degrees} = \frac{7\pi}{36} \times \frac{180}{\pi} = \frac{7 \times 180}{36} = 35 \,\text{degrees} \,. \] Thus, the angle in degrees is \( 35 \,\text{degrees} \).

Step 2: Determine the Reference Angle in Radians

The reference angle is the smallest angle the terminal side of the given angle makes with the \(x\)-axis. Since \( \frac{7\pi}{36} \,\) is less than \( \pi\), it lies in the first quadrant. Hence, the reference angle is the same as the angle itself. Therefore, the reference angle is \( \frac{7\pi}{36} \,\) radians.

Step 3: Drawing the Angle in Standard Position

To draw the angle \( \frac{7\pi}{36} \,\) in standard position:1. Begin by drawing the initial side of the angle along the positive \(x\)-axis.2. Since \( \frac{7\pi}{36} \,\) equates to 35 degrees, rotate the terminal side 35 degrees counterclockwise from the positive \(x\)-axis.3. Label the angle measure on this position.The reference angle, being equal to the original angle in this case, also measures 35 degrees and is visually represented on the same point of rotation.

Key Concepts

Reference Angle

The reference angle is an important concept to understand when working with angles in trigonometry. It is the acute angle formed between the terminal side of the given angle and the x-axis. An acute angle is any angle that is less than 90 degrees or \( \frac{\pi}{2} \) radians. The reference angle is always positive and will always be between 0 and \( \frac{\pi}{2} \) radians.

In the given exercise, where the angle's radian measure is \( \frac{7\pi}{36} \), since the angle is in the first quadrant, the reference angle is the same as the angle itself.

• If the angle is in the first quadrant, the reference angle is simply the same as the original angle.
• In the second quadrant, the reference angle would be \( \pi - \text{angle} \).
• For the third quadrant, calculate the reference angle using \( \text{angle} - \pi \).
• In the fourth quadrant, find it by using \( 2\pi - \text{angle} \).

Understanding reference angles is crucial for simplifying trigonometric functions and for finding angles in different quadrants. It helps in converting any angle back to a relatable and simpler acute angle.

Standard Position

Angles in the "standard position" are defined with their vertex at the origin of a coordinate plane and their initial side on the positive x-axis. To visualize and draw angles effectively, it is important to comprehend the standard position. This position is widely used in mathematics to help explain and calculate trigonometric values and for understanding the rotation of angles.When the given angle, \( \frac{7\pi}{36} \), is drawn in standard position:

• Start with the initial side on the positive x-axis.
• Rotate it counterclockwise if the angle is positive, or clockwise if it is negative.

In the particular case of \( \frac{7\pi}{36} \), the angle is positive, meaning a counterclockwise rotation from the positive x-axis.Understanding angles in this setup allows for calculating and visualizing trigonometric functions, helping in identifying co-terminal angles, and assists in understanding various geometric transformations. It creates a uniform way to handle and analyze angles.

Angle Measurement

In mathematics, angles can be measured in degrees or radians. Understanding both units of measurement is crucial, especially when converting between them.

• Degrees are based on dividing a circle into 360 equal parts, with each part being one degree.
• Radians are a more direct measure, with the angle subtended by an arc equal to the radius of the circle being one radian. A full circle (360 degrees) is equivalent to \(2\pi\) radians.

To convert radians to degrees, use the formula:\[\theta_{degrees} = \theta_{radians} \times \frac{180}{\pi}.\]

For the angle \( \frac{7\pi}{36} \), multiplying by \( \frac{180}{\pi} \) gives 35 degrees. This is a practical example of how to convert radian measure to degree measure. Knowing how to convert between these units is essential for solving problems in trigonometry, physics, and engineering where both measurements may be used interchangeably.

Radian Measure

Radian measure is a way to express angles based on the radius of a circle. One radian is the angle made when the arc length equals the circle's radius. This creates a simple and natural way to measure angles in mathematics.Using radians is particularly advantageous because they provide a direct relationship between the angles and their trigonometric functions. For a circle's full rotation (coming back to the original position), we refer to \( 2\pi \) radians because the circumference of a circle is \( 2\pi \) when the radius is 1.To understand radian measurement better:

• Less than one radian represents an angle that is smaller than the arc it subtends when the arc is less than the radius of the circle.
• If an angle is exactly one radian, it means its subtended arc is exactly equal to the radius.

While working with angles such as \( \frac{7\pi}{36} \), knowing how many radians comprise part of a circle helps us understand the angle's openness.