Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. See Example 6. 1 cos θ = - — 2

Verified step by step guidance

1. Recognize that the equation is \( \cos \theta = -\frac{1}{2} \). We need to find all angles \( \theta \) in the interval \( [0^\circ, 360^\circ) \) where the cosine value is \( -\frac{1}{2} \).
2. Recall the unit circle values where \( \cos \theta = \frac{1}{2} \). These occur at \( \theta = 60^\circ \) and \( \theta = 300^\circ \). Since we want \( \cos \theta = -\frac{1}{2} \), we look for angles where cosine is negative, which happens in the second and third quadrants.
3. Use the reference angle \( 60^\circ \) (since \( \cos 60^\circ = \frac{1}{2} \)) to find the angles in the second and third quadrants where cosine is negative. These angles are \( 180^\circ - 60^\circ \) and \( 180^\circ + 60^\circ \).
4. Calculate the two angles explicitly: \( 180^\circ - 60^\circ = 120^\circ \) and \( 180^\circ + 60^\circ = 240^\circ \). These are the solutions within the given interval.
5. Verify that both angles satisfy the original equation \( \cos \theta = -\frac{1}{2} \) and confirm that they lie within the interval \( [0^\circ, 360^\circ) \).

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 around this circle, with 0° at the positive x-axis. Understanding the unit circle helps identify the coordinates corresponding to cosine and sine values for any angle θ.

Cosine Function and Its Values

The cosine of an angle θ corresponds to the x-coordinate of the point on the unit circle at that angle. Cosine values range from -1 to 1. Knowing that cos θ = -1/2 means θ corresponds to points where the x-coordinate is -0.5, which occurs in specific quadrants of the unit circle.

Solving Trigonometric Equations in a Given Interval

To find all θ in [0°, 360°) satisfying cos θ = -1/2, one must identify all angles within one full rotation where the cosine equals -1/2. This involves recognizing the reference angle and determining the correct quadrants (II and III) where cosine is negative, then expressing the solutions accordingly.