Problem 48 - 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. $$-5^{\circ}, 265^{\circ}$$

Step-by-Step Solution

Answer

-5° is in the fourth quadrant, and 265° is a quadrantal angle on the y-axis.

Step 1: Understanding the Angle -5°

Angles are measured counterclockwise from the positive x-axis in standard position. If the angle is negative, as in -5°, it means we measure 5° in the clockwise direction from the positive x-axis.

Step 2: Identify the Quadrant for -5°

Since -5° is close to the positive x-axis but slightly in the clockwise direction, it lies just below the positive x-axis in the fourth quadrant.

Step 3: Understanding the Angle 265°

For positive angles, we measure counterclockwise from the positive x-axis. The angle 265° means we rotate 265° counterclockwise from the positive x-axis.

Step 4: Identify the Quadrant for 265°

A full circle is 360°, so 265° is less than a full circle. It surpasses 180° (the negative x-axis) and can be further calculated as 265° - 180° = 85°. This means the angle has passed the third quadrant and lies in the fourth quadrant.

Key Concepts

Angles in Standard Position

In trigonometry, understanding angles in standard position is crucial for analyzing their properties and applications. An angle is said to be in standard position when its vertex is at the origin (0,0) of a coordinate plane, and its initial side lies along the positive x-axis. When you measure angles, you usually do so in the counterclockwise direction starting from the positive x-axis. This helps in providing a universal reference point for measuring angles.

• If the angle is positive, you measure it counterclockwise from the positive x-axis.
• For negative angles, you move clockwise from the positive x-axis.

Consider an angle of -5°. It starts from the positive x-axis, and since it's negative, we measure it clockwise. Hence, understanding movement directions is key to identifying where the angle lies on the coordinate plane.
Another example, an angle of 265°, is measured counterclockwise from the positive x-axis, indicating the significant amount of rotation that surpasses both the first and second quadrants.

Quadrantal Angles

Quadrantal angles are special cases in trigonometry. They occur when the terminal side of an angle in standard position aligns with one of the x or y axes. These angles can greatly simplify trigonometric calculations since their terminal sides fall along fixed positions.

• Common quadrantal angles include 0°, 90°, 180°, 270°, and 360°.

For example, an angle such as 90° has its terminal side along the positive y-axis. Quadrantal angles do not reside within any particular quadrant since they lie along the axes. This makes them unique and important when considering angle measurement and trigonometric functions. Although -5° and 265° are not quadrantal angles, understanding these allows clear differentiation from angles lying within quadrants.

Quadrant Identification

Identifying the quadrant in which the terminal side of an angle lies is essential in trigonometry. The coordinate plane is divided into four quadrants

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

When determining the quadrant, consider the direction and size of the angle. For an angle like -5°, it moves slightly clockwise from the positive x-axis, placing it in Quadrant IV.
An angle of 265° requires a significant counterclockwise rotation and surpasses the first three quadrants, also landing in Quadrant IV.
Recognizing the quadrant helps when dealing with trigonometric functions as it affects their signs, for instance, in Quadrant IV, only cosine is positive.