Problem 46 - Algebra and Trigonometry

Question

find the reference angle for each angle. $$ -250^{\circ} $$

Step-by-Step Solution

Answer

The reference angle for \(-250^{\circ}\) is \(70^{\circ}\).

Step 1: Determine the standard position angle

The standard position angle of \(-250^{\circ}\) can be calculated by adding 360 degrees to the given angle since it's negative: \(-250^{\circ} + 360^{\circ} = 110^{\circ}\). Here, 360 degrees are added because it is the measure of a full angle.

Step 2: Determine the reference angle

Since 110 degrees is in the second quadrant (as 90 degrees < 110 degrees < 180 degrees), subtract it from 180 degrees to find the reference angle: \(180^{\circ} - 110^{\circ} = 70^{\circ}\). Hence, the reference angle of \(-250^{\circ}\) is \(70^{\circ}\).

Key Concepts

Standard Position Angle

In trigonometry, the concept of a standard position angle is fundamental to understanding angles and their measurements. A standard position angle is an angle that is measured from the positive x-axis, going counter-clockwise around the coordinate plane's origin. It provides a reference point for measuring the rotation of an angle.

Every angle in trigonometry can be expressed in standard position, even if it's negative or exceeds a full rotation of 360 degrees. For negative angles, we can find their equivalent positive angle in standard position by adding 360 degrees until the result is positive. For example, in the exercise given, the angle of -250 degrees was converted to a standard position angle by adding 360 degrees, resulting in 110 degrees. This process simplifies the understanding of the angle in relation to the coordinate plane.

Trigonometry Quadrants

The coordinate plane in trigonometry is divided into four quadrants, each of which plays a pivotal role in determining the signs of trigonometric functions and the reference angles. The quadrants are named in a counter-clockwise direction starting from the upper right quadrant:

• Quadrant I (0 to 90 degrees)
• Quadrant II (90 to 180 degrees)
• Quadrant III (180 to 270 degrees)
• Quadrant IV (270 to 360 degrees)

Angles falling into different quadrants will have different ranges, and those specific ranges are handy when evaluating trigonometric functions, such as sine, cosine, and tangent. Each quadrant has a unique combination of positive and negative values for these functions. For instance, in the provided exercise, the angle of 110 degrees falls in Quadrant II, where sine is positive and cosine and tangent are negative.

Calculating Reference Angles

A reference angle is the smallest angle formed between the terminal side of an angle in standard position and the x-axis. The reference angle is always positive and is never larger than 90 degrees.

To calculate reference angles, one must first determine the angle's quadrant positioning. Once that is known, we can calculate the reference angle using specific formulas for each quadrant. For angles in Quadrant II, like our example of 110 degrees, the reference angle is found by subtracting the angle from 180 degrees, resulting in a reference angle of 70 degrees. The purpose of reference angles is to simplify complex angle measurements and make it easier to use trigonometric functions by correlating them with an acute angle that shares the same sine, cosine, and tangent values (apart from the sign).

Negative Angles in Trigonometry

In trigonometry, it's essential to understand that angles can be positive or negative based on their direction of rotation from the initial side to the terminal side. Positive angles rotate counter-clockwise, while negative angles rotate clockwise.

Negative angles are often encountered in trigonometric equations and problems. To work with them effectively, we convert negative angles to their positive counterparts by adding or subtracting full rotations (multiples of 360 degrees) until we arrive at a positive standard position angle. For example, the given angle of -250 degrees, when adjusted by adding 360 degrees, gives us the positive standard position angle of 110 degrees. This allows for consistency when assessing the angle's quadrant and when calculating functions and reference angles, ensuring clarity in trigonometric problem-solving.