Problem 11 - Algebra 2

Question

\(\cos \theta=-\frac{1}{2},\) Quadrant II

Step-by-Step Solution

Answer

The angle is \(120^\circ\) or \(\frac{2\pi}{3}\) radians.

Step 1: Identify the Reference Angle

Since \( \cos \theta = -\frac{1}{2} \) in Quadrant II, the cosine value is negative. We look for the reference angle where the cosine is \( \frac{1}{2} \). This is a standard angle from trigonometry, \( \cos(\theta) = \frac{1}{2} \) corresponds to \( \theta = 60^\circ \) or \( \theta = \frac{\pi}{3} \) radians.

Step 2: Determine the Quadrant II Angle

In Quadrant II, the cosine is negative and angles are computed as \( 180^\circ - \alpha \) or \( \pi - \alpha \), where \( \alpha \) is the reference angle. Given our reference angle \( \alpha = 60^\circ \) or \( \frac{\pi}{3} \), the angle in Quadrant II is \( 180^\circ - 60^\circ = 120^\circ \) or \( \pi - \frac{\pi}{3} = \frac{2\pi}{3} \) radians.

Key Concepts

Cosine function

The cosine function is a fundamental part of trigonometry. It helps us understand relationships between angles and sides in right-angled triangles.

In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. Mathematically, it is expressed as:

• \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}

Cosine values range from -1 to 1. When using the unit circle, the cosine of an angle corresponds to the x-coordinate of a point on the circle. In different quadrants of the Cartesian plane, the cosine function takes different signs. For example, in Quadrant II, the cosine values are negative, as the x-coordinate here is negative.

This function is periodic, meaning it repeats its values in a regular pattern over 360° or \(2\pi\) radians. Understanding the cosine function helps us solve problems involving angles and can even be applied to real-world scenarios like wave patterns and oscillations.

Reference angle

A reference angle is the smallest angle that a given angle forms with the horizontal axis.

It is always positive, acute, and essential in determining the values of trigonometric functions for other angles. No matter which quadrant the original angle resides in, the reference angle is always between 0° and 90°, or between 0 and \(\frac{\pi}{2}\) radians.

• To find the reference angle in degrees, take the absolute difference between the given angle and the nearest x-axis (0° or 180°).
• In radians, subtract from \(\pi\) or \(0\) accordingly.

Understanding reference angles simplifies calculations, as it allows us to use known trigonometric values of common angles, such as 30°, 45°, or 60°, regardless of the original angle's placement in the coordinate system.

These angles help us find equivalent values when applying the symmetry of the trigonometric circle, making them highly useful in solving complex problems.

Quadrant II

Quadrant II in the Cartesian coordinate system is the upper left section of the plane.

Here, angles range from 90° to 180° or from \(\frac{\pi}{2}\) to \(\pi\) radians.

This quadrant is unique because cosine values are negative while sine values remain positive. The negative cosine indicates that the x-coordinates in this region are below the horizontal axis.

• Angles in Quadrant II are calculated as \(180^\circ - \alpha\) for degrees and \(\pi - \alpha\) for radians, where \(\alpha\) represents the reference angle.

This characteristic enables us to use reference angles to find precise trigonometric values for specific angles in this quadrant. For example, if \(\theta\) in Quadrant II has a reference angle of 60°, then \(\theta = 120^\circ\) or \(\theta = \frac{2\pi}{3}\) radians.

The ability to find and use these angles is critical in solving trigonometric equations and understanding the behavior of trigonometric functions across different sections of the unit circle.