Problem 22 - Precalculus : Building Concepts and Connections

Question

Skills This set of exercises will reinforce the skills illustrated in this section. In Exercises \(9-22,\) find the reference angle for each of the angles given. $$-135^{\circ}$$

Step-by-Step Solution

Answer

The reference angle for \(-135^{\circ}\) is \(45^{\circ}\)

Step 1: Determine the quadrant of the given angle

Angles are measured in standard position, starting from the positive x-axis and moving counterclockwise for positive angles and clockwise for negative angles. \(-135^{\circ}\) places the angle in the second quadrant because it has moved 135 degrees clockwise from the positive x-axis.

Step 2: Calculate the reference angle

The reference angle is found by calculating the smallest positive angle to the x-axis. For an angle in the second quadrant, the reference angle \(\theta_{r}\) is found by subtracting the angle from \(180^{\circ}\) as \(180^{\circ} - \text{angle}\). Thus, \(\theta_{r} = 180^{\circ} - (-135^{\circ}) = 315^{\circ}\)

Step 3: Correct the reference angle if necessary

The reference angle should be between \(0^{\circ}\) and \(90^{\circ}\). Since the calculated angle is out of this range, subtract it from \(360^{\circ}\) to find the correct reference angle. Thus, \(\theta_{r} = 360^{\circ} - 315^{\circ} = 45^{\circ}\)

Key Concepts

Trigonometric Angles

Trigonometric angles are an essential part of understanding how angles relate to one another in a circle, specifically a unit circle. An angle in trigonometry can be measured in degrees or radians. A full circle equals 360 degrees or \(2\pi\) radians. Trigonometric angles are typically measured from a fixed beginning line — the positive x-axis — and move in a counterclockwise direction for positive angles. If the angle is negative, it moves clockwise from the positive x-axis.

Understanding trigonometric angles is crucial because it lays the foundation for calculations involving trigonometric functions like sine, cosine, and tangent. Remember that each angle can be associated with a specific point on the unit circle, which helps in visualizing and calculating angles that are in non-standard positions.

Quadrants in Trigonometry

The concept of quadrants in trigonometry refers to the different sections of the xy-plane divided by the x and y axes. These quadrants help in identifying the sign of trigonometric functions for specific angles.

• First Quadrant: where both x and y are positive. Angles here range from 0° to 90°.
• Second Quadrant: where x is negative and y is positive. Angles here range from 90° to 180°.
• Third Quadrant: where both x and y are negative. Angles here range from 180° to 270°.
• Fourth Quadrant: where x is positive and y is negative. Angles range from 270° to 360°.

Knowing which quadrant an angle is in helps to determine the sign of the trigonometric function values. For instance, sine is positive in the first and second quadrants, while cosine is only positive in the first and fourth quadrants.

Negative Angles

Negative angles allow us to move in the clockwise direction around the unit circle. It is essential in trigonometry, as it provides a way to calculate angles that are not in the standard positive direction. These angles are still measured from the positive x-axis, but they decrease as they move clockwise.

For example, an angle of \(-135^{\circ}\) means you move 135 degrees in a clockwise direction starting from the positive x-axis. The idea of negative angles complements positive angles by offering a more flexible understanding of movement around a circle. In practical terms, it helps determine the same final point on the circle as with positive angles but by taking a different path. Understanding negative angles is key to find reference angles and calculate precise trigonometric function values.