Problem 47 - Basic Technical Mathematics with Calculus

Question

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° lies in the First Quadrant; -270° is a Quadrantal Angle.

Step 1: Understand the problem

We need to determine the quadrant in which the terminal side of each angle lies or identify it as a quadrantal angle. A quadrantal angle means the terminal side lies on the x-axis or y-axis.

Step 2: Normalize the angle for 435°

For an angle greater than 360°, subtract multiples of 360° to find an equivalent angle. Subtract 360° from 435°: \( 435° - 360° = 75° \).

Step 3: Determine the quadrant for 75°

An angle of 75° is less than 90° and more than 0°, meaning it lies in the **First Quadrant**.

Step 4: Normalize the angle for -270°

For a negative angle, add multiples of 360° until the angle is positive within the range from 0° to 360°. Add 360° to -270°: \( -270° + 360° = 90° \).

Step 5: Determine the position for 90°

An angle of 90° is exactly on the positive y-axis, which makes it a **Quadrantal Angle**.

Key Concepts

Angular Measurement

When studying trigonometry, understanding angular measurement is crucial. Angles are measured in degrees or radians. Degrees are a common unit that divides a full circle into 360 equal parts. Each degree is further subdivided into 60 minutes, much like time. To find the equivalent of an angle larger than 360°, you subtract multiples of 360° until it falls within 0° to 360°, like how we did for the angle 435°. It reduces to 75° by subtracting 360°. Similarly, for negative angles, which occur when you measure an angle in the clockwise direction, you add multiples of 360° to bring them into the 0° to 360° range. This is why -270° becomes 90° when 360° is added.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four sections known as quadrants. Each quadrant represents a 90-degree segment of the circle:

• **First Quadrant:** Angles range from 0° to 90°. Both sine and cosine values are positive.
• **Second Quadrant:** Covers angles from 90° to 180°, where sine values are positive but cosine values are negative.
• **Third Quadrant:** Spanning 180° to 270°, both sine and cosine are negative.
• **Fourth Quadrant:** Ranges from 270° to 360°, with positive cosine and negative sine values.

For each quadrant, knowing the signs of trigonometric functions helps solve problems efficiently. In our example, 75° is in the First Quadrant, where both sine and cosine are positive.

Positive and Negative Angles

Angles can be positive or negative based on their direction of measurement. A positive angle is created by rotating counterclockwise from the initial side, while a negative angle is formed by a clockwise rotation. To interpret angles, we need to adjust them to be within 0° to 360° using the methods mentioned before. This normalization helps in identifying the quadrant or position on the axes. An example is -270°, which was converted to 90°, a quadrantal angle that lies on the positive y-axis, often signaling the change or transition between quadrants. By understanding the nature of positive and negative angles, you can tackle more complex trigonometry problems with greater confidence.