Find all angles, 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 
∘
 ≤θ<360 
∘
 , that solve the following equation.
cosine, theta, equals, minus, one half


We are asked to find all angles \(\theta\) such that:

\[
\cos(\theta) = -\frac{1}{2}
\]

and \(0^\circ \leq \theta < 360^\circ\).

### Step 1: Reference angle
The cosine value of \(-\frac{1}{2}\) is negative, so we are looking for angles in the second and third quadrants because cosine is negative in these two quadrants.

The reference angle is the angle for which \(\cos(\theta) = \frac{1}{2}\). The angle corresponding to \(\cos^{-1}\left(\frac{1}{2}\right)\) is \(60^\circ\).

### Step 2: Find angles in the second and third quadrants
In the second quadrant, the angle is given by:

\[
\theta_1 = 180^\circ - 60^\circ = 120^\circ
\]

In the third quadrant, the angle is given by:

\[
\theta_2 = 180^\circ + 60^\circ = 240^\circ
\]

### Step 3: Final answer
The angles that satisfy \(\cos(\theta) = -\frac{1}{2}\) in the interval \(0^\circ \leq \theta < 360^\circ\) are:

\[
\theta = 120^\circ, 240^\circ
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do we find reference angles for trigonometric functions?
2. Why is cosine negative in the second and third quadrants?
3. What is the general solution for \(\cos(\theta) = -\frac{1}{2}\)?
4. Can you solve \(\cos(\theta) = \frac{1}{2}\) in the same interval?
5. How does the unit circle help in solving trigonometric equations?

**Tip:** Always check in which quadrants the trigonometric function is positive or negative when solving for angles.

Find all angles, 0° ≤ θ < 360°, that solve the equation cos(θ) = -1/2.

Learn how to find the angles θ that solve the trigonometric equation cos(θ) = -1/2 in the range 0° ≤ θ < 360°. This problem uses concepts of trigonometry, including reference angles and the cosine function's behavior in different quadrants. Suitable for students in Grades 9-11.

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