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Magazine Chapter 13: Problem 39
Sketch each angle. Then find its reference angle. \(-125^{\circ}\)
Short Answer
Expert verified
The reference angle is \(55^{\circ}\).
Step by step solution
01
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.
02
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.
03
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 \(235^{\circ} - 180^{\circ} = 55^{\circ}\).
04
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 \(55^{\circ}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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掳) that helps reference the original angle.
To calculate the reference angle, consider the following:
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掳.
- In the third quadrant, subtract 180掳 from the angle.
- For angles in the fourth quadrant, subtract the angle from 360掳.
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掳 to 90掳) - All sine, cosine, and tangent values are positive.
- Second Quadrant (90掳 to 180掳) - Sine values are positive while cosine and tangent are negative.
- Third Quadrant (180掳 to 270掳) - Tangent values are positive, but sine and cosine are negative.
- Fourth Quadrant (270掳 to 360掳) - Cosine values are positive, but sine and tangent are 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:
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.
