Problem 47 - Basic Technical Mathematics with Calculus

Question

The given angles are in standard position. Designate each angle by the quadrant in which the terminal side lies, or as a quadrantal angle. $$435^{\circ},-270^{\circ}$$

Step-by-Step Solution

Answer

435° is in the first quadrant; -270° is a quadrantal angle on the positive y-axis.

Step 1: Understand the Problem

We need to determine the quadrant location of two given angles: \(435^{\circ}\) and \(-270^{\circ}\). Each angle in standard position has its initial side on the positive x-axis.

Step 2: Angle Reduction (435°)

Since angles are periodic every \(360^{\circ}\), we can reduce \(435^{\circ}\) by subtracting \(360^{\circ}\) until it lies within the principal range \([0^{\circ}, 360^{\circ})\). \[435^{\circ} - 360^{\circ} = 75^{\circ}\] Thus, \(435^{\circ} = 75^{\circ}\) in standard position.

Step 3: Determine Quadrant (435°)

An angle of \(75^{\circ}\) lies in the first quadrant because it is between \(0^{\circ}\) and \(90^{\circ}\).

Step 4: Angle Reduction (-270°)

For negative angles, we add \(360^{\circ}\) until it lies within \([0^{\circ}, 360^{\circ})\). \[-270^{\circ} + 360^{\circ} = 90^{\circ}\] Thus, \(-270^{\circ} = 90^{\circ}\) in standard position.

Step 5: Determine Quadrant (-270°)

An angle of \(90^{\circ}\) is a quadrantal angle since it lies on the positive y-axis. A quadrantal angle is not in any quadrant but on the axis.

Key Concepts

Angle Reduction

When dealing with angles in trigonometry, one common practice is angle reduction. This means simplifying an angle to fall within a specific range, usually between \(0^{\circ}\) and \(360^{\circ}\).

• **Positive Angle Reduction:** If an angle is greater than \(360^{\circ}\), like \(435^{\circ}\), subtract \(360^{\circ}\) as many times as needed to bring it into the principal range.
• **Negative Angle Reduction:** Conversely, for negative angles like \(-270^{\circ}\), add \(360^{\circ}\). Keep doing this until the angle lies within the range of \([0^{\circ}, 360^{\circ})\).

By reducing \(435^{\circ}\), for example, we find it is equivalent to \(75^{\circ}\).This concept helps in quickly identifying properties like the quadrant in which the angle's terminal side lies. For \(-270^{\circ}\), reducing it gives us \(90^{\circ}\).Knowing how to perform angle reduction is crucial for solving problems involving angles given in standard position.

Quadrantal Angles

Quadrantal angles are unique in trigonometry because their terminal sides lie directly on one of the axes. They do not fall into any quadrant.

• These angles include \(0^{\circ}\), \(90^{\circ}\), \(180^{\circ}\), \(270^{\circ}\), and \(360^{\circ}\) (which coincides with \(0^{\circ}\)).
• Quadrantal angles are important because they mark the boundaries of quadrants.
• Trigonometric values (like sine and cosine) have specific, easily recognizable values at these angles, often \(0\), \(1\), or \(-1\).

For instance, when we reduced \(-270^{\circ}\), it became \(90^{\circ}\), a classic quadrantal angle lying on the positive y-axis.Understanding quadrantal angles helps in visualizing angle positions, especially when considering their trigonometric values.

Standard Position Angles

A standard position angle is an angle placed on a coordinate plane. This positioning is key in trigonometry as it sets a common starting point.

• The angle's vertex is at the origin \((0,0)\) of the coordinate plane.
• The initial side of the angle is along the positive x-axis.
• The terminal side is where the angle "ends," and where it helps determine which quadrant the angle occupies.

In identifying the quadrant, angles are measured counter-clockwise (positive) from the initial side. Thus, a \(75^{\circ}\) angle ends in the first quadrant, while a reduced form of \(-270^{\circ}\), \(90^{\circ}\), rests on the y-axis as a quadrantal angle.Utilizing standard position angles is helpful for consistency and simplifies calculations involving trigonometric functions.