Problem 47 - Algebra and Trigonometry

Question

find the reference angle for each angle. $$ -335^{\circ} $$

Step-by-Step Solution

Answer

The reference angle for -335° is 25°.

Step 1: Understanding Reference Angle

The reference angle is the acute version of the given angle. This angle is always positive and is formed between the terminal side of the given angle and the x-axis. It's always less than 90 degree or equal to 90 degree.

Step 2: Determine Quadrant

The given angle is -335 deg. As the angle is negative, it is rotating in a clockwise direction, so this position is in the first quadrant.

Step 3: Calculate Reference Angle

Subtract the rotation counterclockwise from the rotating angle. Here, since it is a first quadrant angle, its reference angle is simply 360° + (-335°).\nThe reference angle is simply itself or \(360° - \|-335°\|\) = 25°

Key Concepts

Terminal Side

The terminal side of an angle is a fundamental concept in trigonometry. Imagine a line that rotates around a fixed point known as the origin in the coordinate system. The starting position of this line is called the initial side. When the line stops rotating, the end position is known as the terminal side. Here's what you need to know about the terminal side:

• It helps in determining the angle's position in the coordinate plane.
• The angle is measured from the initial side to the terminal side.

Using a coordinate plane makes understanding the terminal side clearer. Remember that the terminal side can lie in any of the four quadrants.

Clockwise Direction

Angles can be measured in two directions: clockwise and counterclockwise. Whether an angle is positive or negative depends on its direction.

• Clockwise direction refers to moving in the same direction as the hands of a clock.
• Negative angles are those measured in a clockwise direction.

In our problem, when we move -335°, it means we start from the initial side and rotate 335° in a clockwise direction.
This concept is crucial as it defines the way we perceive and calculate angles in geometry.

First Quadrant

The first quadrant is one of the four sections of the coordinate plane. Starting from the x-axis, it is the top-right section of the plane. Here's why it matters:

• Angles in the first quadrant are generally positive presumed from 0° to 90°.
• Both x and y coordinates of points in the first quadrant are positive.

In our example, the angle -335° implies a clockwise movement that is interpreted to end up in the first quadrant.
Understanding which quadrant an angle is in helps predict its characteristics and reference angles.

Acute Angle

An acute angle is an angle that is less than 90°. It is one of the primary types of angles in trigonometry.

• Acute angles are always positive.
• They form by the terminal side and the x-axis, usually residing in the first quadrant.

In the calculation for a reference angle, ensuring the result is an acute angle (25° in our exercise) helps in mapping the angle's actual position on the coordinate plane.
Acute angles are foundational in geometry, playing a role in creating points of reference for other angles.