Problem 11 - Technical Mathematics with Calculus

Question

If \(\theta\) is an angle in standard position, state in what quadrants its terminal side can lie if $$\theta=-415^{\circ}.$$

Step-by-Step Solution

Answer

The terminal side of the angle \(\theta = -415^\circ\) lies in the fourth quadrant.

Step 1: Understand the Angle Measurement

The given angle, \(\theta = -415^\circ\), is negative, indicating that it is measured in the clockwise direction from the positive x-axis.

Step 2: Convert Negative Angle to Positive Equivalent

To determine where the terminal side of the angle lies, add full rotations of 360 degrees until the resulting angle is positive. One full rotation is 360 degrees, so add this to -415 degrees: \(\theta_{pos} = -415^\circ + 360^\circ = -55^\circ\).

Step 3: Adjust Angle to Fall Within One Rotation

Since \(\theta_{pos}\) is still negative, add another 360 degrees to bring it to the equivalent positive angle: \(\theta_{equiv} = -55^\circ + 360^\circ = 305^\circ\).

Step 4: Determine the Quadrant of the Terminal Side

Angles between 270 and 360 degrees terminate in the fourth quadrant. Therefore, for the given angle of \(305^\circ\), its terminal side will lie in the fourth quadrant.

Key Concepts

Standard Position of an Angle

In mathematics, particularly in trigonometry and geometry, the standard position of an angle refers to an angle with its vertex at the origin of the coordinate plane and its initial side aligned with the positive x-axis. When drawing or measuring angles, the standard position provides a reference point that ensures consistency and helps us accurately determine where the terminal side, or the 'ending side' of the angle, falls.

To visualize this, imagine a line that starts at the center of a coordinate plane and extends to the right, lying along the positive x-axis. This is where all angles begin in standard position. If the angle is positive, we rotate this line counterclockwise; for negative angles, it rotates clockwise.

Negative Angles

Understanding negative angles is crucial for solving many problems in trigonometry. As opposed to positive angles, which are measured counterclockwise, negative angles are measured clockwise from the positive x-axis. This means that if you have an angle such as \( -415^\circ \), it tells you to rotate clockwise from the standard starting point on the positive x-axis.

However, measuring angles does not stop at \( -360^\circ \) or \( 360^\circ \); angles can be far greater or less. Whenever you come across angles that aren't between \( 0^\circ \) and \( 360^\circ \), you can usually simplify them by adding or subtracting full rotations of \( 360^\circ \) until you get an equivalent angle that falls within this range. This process is known as 'angle normalization,' and it simplifies the angle to find its position in the coordinate plane.

Quadrants of the Coordinate System

The quadrants of the coordinate system are essential for locating points, plotting graphs, and understanding angles in trigonometry. The coordinate plane is divided into four areas, called quadrants, by the x and y axes, which intersect at the origin.

These quadrants are numbered from I to IV and are as follows:

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y coordinates are negative.
• Quadrant IV: x is positive, y is negative.

When determining the quadrant where an angle's terminal side lies, focus on its measure in the standard position after it has been normalized. For instance, an angle with a measure of \( 305^\circ \) terminates in Quadrant IV because it is more than \( 270^\circ \) but less than \( 360^\circ \). Understanding how to link angle measures to quadrants is a key skill in trigonometry.