Problem 39 - Algebra 2

Question

Sketch each angle. Then find its reference angle. \(-125^{\circ}\)

Step-by-Step Solution

Answer

The reference angle is \(55^{\circ}\).

Step 1: Identify the Quadrant

The angle given is \(-125^{\circ}\). Since it is negative, it is measured clockwise from the positive x-axis. Start by adding \(360^{\circ}\) to find the equivalent positive angle: \(-125^{\circ} + 360^{\circ} = 235^{\circ}\). The angle of \(235^{\circ}\) lies in the third quadrant.

Step 2: Sketch the Angle

To sketch \(-125^{\circ}\), move clockwise from the positive x-axis. Initially, \(-90^{\circ}\) gets you to the negative y-axis (downward), and continuing to \(-125^{\circ}\) places the terminal side in the third quadrant, closer to the negative x-axis.

Step 3: Calculate the Reference Angle

The reference angle is the acute angle that the terminal side of the angle makes with the x-axis. For angles in the third quadrant, subtract \(180^{\circ}\) from the angle. The reference angle is \(235^{\circ} - 180^{\circ} = 55^{\circ}\).

Step 4: Verification

Verify the calculations: Since the angle \(235^{\circ}\) was in the third quadrant, and subtracting \(180^{\circ}\) gives a positive angle, the reference angle for \(-125^{\circ}\) is indeed \(55^{\circ}\).

Key Concepts

Reference Angle

In trigonometry, a reference angle is an important concept for simplifying the understanding and calculation of an angle. The reference angle is always the smallest angle a given angle makes with the x-axis. The goal is to find an acute angle (an angle less than 90°) that helps reference the original angle.

To calculate the reference angle, consider the following:

• For angles in the first quadrant, the reference angle is the angle itself.
• In the second quadrant, subtract the angle from 180°.
• In the third quadrant, subtract 180° from the angle.
• For angles in the fourth quadrant, subtract the angle from 360°.

For example, if you have an angle of 235^ { ext{°}, as in our problem, it's located in the third quadrant. Therefore, the reference angle is 235° - 180° = 55°, making it easy to work with trigonometric functions that rely on acute angles.

Quadrant Determination

Understanding which quadrant an angle lies in is foundational to mastering trigonometry. Angles provide direction from the positive x-axis, and are measured counterclockwise. However, negative angles advance clockwise, making them equally important in calculations. Quadrants in a coordinate plane are defined in the following order:

• First Quadrant (0° to 90°) - All sine, cosine, and tangent values are positive.
• Second Quadrant (90° to 180°) - Sine values are positive while cosine and tangent are negative.
• Third Quadrant (180° to 270°) - Tangent values are positive, but sine and cosine are negative.
• Fourth Quadrant (270° to 360°) - Cosine values are positive, but sine and tangent are negative.

For example, with our angle of -125°, converting it to 235° places it in the third quadrant. This means while calculating trigonometric functions from this angle, we should expect the tangent to be positive, but sine and cosine to be negative.

Angle Sketching

Visualizing angles helps reinforce understanding and assists in calculations. Sketching an angle involves showing the angle's direction from a starting position (the positive x-axis) and indicating its terminal side. When dealing with negative angles, begin by moving clockwise from the positive x-axis.

Below is a simple method to sketch an angle:

• Identify if the angle is positive or negative.
• If positive, proceed counterclockwise from the positive x-axis. If negative, move clockwise.
• Stop at the degree measurement of your angle.

For our angle -125°, moving 125° clockwise lands us in the third quadrant. Starting by moving -90° positions us along the negative y-axis, and continuing to -125° situates the terminal side near the negative x-axis. This visualization ensures a better grasp of reference angles and placements in relevant trigonometric contexts.