Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. cos θ = -½

Verified step by step guidance

1. Recall that the cosine function corresponds to the x-coordinate on the unit circle for an angle \( \theta \). We are given \( \cos \theta = -\frac{1}{2} \), and we need to find all \( \theta \) in the interval \( [0^\circ, 360^\circ) \) that satisfy this.
2. Identify the reference angle where \( \cos \theta = \frac{1}{2} \). Since cosine is positive in the first quadrant, find the acute angle \( \alpha \) such that \( \cos \alpha = \frac{1}{2} \). This reference angle \( \alpha \) is \( 60^\circ \) because \( \cos 60^\circ = \frac{1}{2} \).
3. Since the cosine value is negative, \( \theta \) must lie in the quadrants where cosine is negative. Cosine is negative in the second and third quadrants.
4. Use the reference angle to find the solutions in the second and third quadrants: - In the second quadrant, \( \theta = 180^\circ - \alpha \). - In the third quadrant, \( \theta = 180^\circ + \alpha \).
5. Write the general solutions explicitly using the reference angle \( 60^\circ \): - \( \theta = 180^\circ - 60^\circ \) - \( \theta = 180^\circ + 60^\circ \) These are the values of \( \theta \) in the interval \( [0^\circ, 360^\circ) \) where \( \cos \theta = -\frac{1}{2} \).

Key Concepts

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles in trigonometry are often measured in degrees or radians on this circle, where the cosine of an angle corresponds to the x-coordinate of the point on the circle. Understanding the unit circle helps identify where cosine values are positive or negative.

Cosine Function and Its Values

The cosine function relates an angle to the ratio of the adjacent side over the hypotenuse in a right triangle, or equivalently, the x-coordinate on the unit circle. Knowing that cos θ = -½ means finding angles where the x-coordinate is -0.5, which occurs in specific quadrants where cosine is negative.

Solving Trigonometric Equations in a Given Interval

To solve cos θ = -½ within [0°, 360°), identify all angles in this range where the cosine equals -½. Since cosine is negative in the second and third quadrants, use reference angles and symmetry properties to find all solutions within the specified interval.