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Chapter 1: Problem 35

Sketch each angle in standard position. (a) \(30^{\circ}\) (b) \(150^{\circ}\)

Short Answer

Expert verified
The first angle \(30^{\circ}\) would fall in the first quadrant, between the positive x and y axes. The second angle \(150^{\circ}\) would fall in the second quadrant, between the positive y-axis and the negative x-axis.

Step by step solution

01

Understanding Unit Circle and Angles

The critical point to understand while converting standard position angles to a sketch is that angles are measured from the positive x-axis, and in the counterclockwise direction. The angle increases as we go counterclockwise on the unit circle. The unit circle is a circle with a radius of one that is centered at the origin of the coordinate plane.
02

Sketching the First Angle of \(30^{\circ}\) in standard position

The first angle is \(30^{\circ}\). To sketch this, start from the positive x-axis (which is set at 0 degrees) and measure \(30^{\circ}\) counterclockwise. As we are moving less than a quarter turn, the terminal side of the angle will be in the first quadrant, somewhere between the positive x and y axes.
03

Sketching the Second Angle of \(150^{\circ}\) in standard position

The second angle given is \(150^{\circ}\). Starting again from the positive x-axis, this time we measure half a circle and then subtract \(30^{\circ}\) because \(180-150=30\). So we move around until we are \(30^{\circ}\) short of the negative x-axis. The terminal side of this angle will fall in the second quadrant, between the positive y-axis and the negative x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Circle
The unit circle is a fundamental concept in trigonometry and helps in understanding angles in standard position. Imagine a circle with a radius of exactly one unit, centered at the origin (0,0) of a coordinate plane. This circle aids in visualizing angles and their trigonometric ratios.
  • Every point on the unit circle corresponds to an angle's position.
  • The x-coordinates of these points represent cosine values, while y-coordinates represent sine values of the angle.
When an angle is in standard position, its initial side lies on the positive x-axis. As the angle increases, its terminal side moves counterclockwise around the circle. Thus, understanding the unit circle helps correlate angle measurements to trigonometric functions and identify where the terminal side of an angle lies.
Degree Measurement
Degree measurement is a way of expressing the size of an angle. A full circle consists of 360 degrees, which makes degrees a familiar unit for measuring angles. This form of measurement is part of the sexagesimal system that divides a circle into 360 equal parts.
  • An angle of 90 degrees is a right angle.
  • 180 degrees marks a straight line.
  • 270 degrees points directly downwards, completing three-quarters of a full circle.
In the context of sketching angles in standard position, degree measurement allows you to determine exactly how far to rotate from the positive x-axis to reach the terminal side of an angle. Measuring angles in degrees provides an accessible way to relate geometric concepts to real-world situations.
First Quadrant
Quadrants are segments that divide the coordinate plane into four regions. The first quadrant is located in the top-right section of the plane. Here, both x and y coordinates are positive. When an angle in standard position terminates in this quadrant, its measure is between 0 and 90 degrees.
  • Angle sizes here range from a straight line (0 degrees) to a vertical line pointing upwards (90 degrees).
  • All trigonometric values (sine, cosine, and tangent) are positive in this region.
The first quadrant is significant because it establishes the foundational understanding of trig ratios and how angles operate in an introductory setting. For example, the angle of 30 degrees, as mentioned in the exercise, falls in the first quadrant.
Second Quadrant
The second quadrant is located in the top-left section of the coordinate plane. In this region, the x-coordinates are negative, but the y-coordinates remain positive. Angles in standard position that terminate here have measures ranging from 90 to 180 degrees.
  • Angles in the second quadrant proceed counterclockwise from a vertical line pointing upwards to a horizontal line pointing leftward.
  • In this quadrant, sine values remain positive, while cosine and tangent values become negative.
Understanding the characteristics of angles in the second quadrant allows you to analyze angles that have extended past the initial quarter turn. For example, an angle of 150 degrees, as in the exercise, lies in this quadrant. This knowledge reveals the multiple cyclical behaviors of angles and enhances comprehension of how they interact with the unit circle.

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