Question

If \(\theta\) is an angle in standard position, state in what quadrants its terminal side can lie if $$\theta=272^{\circ}.$$

Short answer

The terminal side of the angle \theta = 272^{\textrm{o}} lies in the 4th quadrant.

Step-by-step solution

Understanding the Exercise

The exercise asks us to determine in which quadrant the terminal side of the angle in standard position lies when the angle is 272 degrees. Angles in standard position have their initial side on the positive x-axis and their terminal side determined by the angle's measure in degrees, measured counterclockwise.

Determining the Quadrant

Since a circle is 360 degrees, we divide it into four quadrants of 90 degrees each. Starting from 0 degrees on the positive x-axis: 0 to 90 degrees is the 1st quadrant, 90 to 180 degrees is the 2nd quadrant, 180 to 270 degrees is the 3rd quadrant, and 270 to 360 degrees is the 4th quadrant. An angle of 272 degrees falls between 270 and 360 degrees. Therefore, the terminal side of an angle measuring 272 degrees lies in the 4th quadrant.

Key concepts

Standard Position of Angles

The concept of the standard position of angles is fundamental to understanding angle measurement and orientation. An angle is said to be in standard position when its initial side coincides with the positive x-axis of a coordinate plane.

In this context, the vertex of the angle is located at the origin of the coordinate system, which means the point where the x and y-axes intersect. The angle's terminal side is what we actually rotate to create different angle measurements. The rotation is typically in a counter-clockwise direction, implying a positive angle, while a clockwise rotation indicates a negative angle.

It is crucial for students to grasp this concept because it forms the basis for identifying in which quadrant an angle lies. When dealing with angles more than 360 degrees or less than 0 degrees, it is the terminal side after completing full rotations that determines the position, not just the numerical value of the angle.

Measuring Angles

Measuring angles is a process that involves determining the amount of rotation required to get the terminal side from its initial standard position. This rotation is quantified in degrees or radians. One complete rotation around a circle corresponds to 360 degrees or 2π radians.

To measure an angle from the standard position, you begin at 0 degrees (or 0 radians) along the positive x-axis and proceed in a counter-clockwise direction for positive angles or clockwise for negative angles. For instance, a 45-degree angle would fall in the first quadrant, halfway between 0 and 90 degrees.

Angles can also be measured in gradians, where a complete rotation equals 400 gradians. Despite the various units, the principles of angle measurement remain consistent. Understanding how to measure angles correctly is essential for correctly identifying their quadrant location and for solving complex trigonometry problems.

Quadrants of a Circle

The quadrants of a circle are a division of the circular shape into four distinct parts, each containing a range of angle measurements. In the standard position, the positive x-axis is the starting point. The first quadrant is where angle measures from 0 to 90 degrees are found. A step further, the second quadrant contains angles from 90 to 180 degrees.

Moving along, the third quadrant encompasses angles between 180 and 270 degrees, where angles are now positioned below the x-axis. Lastly, the fourth quadrant holds angles from 270 to 360 degrees, completing the full circle before the terminal side returns to the positive x-axis, marking the start of another rotation.

The importance of understanding quadrants is not just for identifying the location of an angle but also has implications for the sign of the trigonometric functions (sine, cosine, and tangent) associated with angles in those quadrants. For example, only cosine and secant are positive in the fourth quadrant where an angle like 272 degrees resides.