Find the degree measure of θ if it exists. Do not use a calculator.
θ = arccos (-1/2)

Verified step by step guidance

Recall that the function \( \arccos(x) \) gives the angle \( \theta \) in the range \( 0^\circ \leq \theta \leq 180^\circ \) whose cosine is \( x \).

Identify the value given inside the arccos function: \( \cos \theta = -\frac{1}{2} \).

Think about the unit circle and the cosine values of common angles. Cosine corresponds to the x-coordinate on the unit circle.

Recall that \( \cos 60^\circ = \frac{1}{2} \), so \( \cos \theta = -\frac{1}{2} \) means \( \theta \) is an angle where the cosine is the negative of \( \frac{1}{2} \).

Since cosine is negative in the second quadrant (between \( 90^\circ \) and \( 180^\circ \)), find the angle in that range whose cosine is \( -\frac{1}{2} \). This angle is \( 180^\circ - 60^\circ \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Trigonometric Functions

Inverse trigonometric functions, like arccos, return the angle whose trigonometric ratio equals a given value. For arccos(x), it gives the angle θ in the range 0° to 180° such that cos(θ) = x. Understanding this helps find the angle from a cosine value without a calculator.

Cosine Values of Special Angles

Certain angles have well-known cosine values, such as 60° where cos(60°) = 1/2. Recognizing these special angles and their cosine values allows you to identify θ when given cos(θ) = -1/2, by considering the unit circle and symmetry.

Unit Circle and Angle Quadrants

The unit circle helps determine the angle corresponding to a cosine value, considering the sign and quadrant. Since cosine is negative in the second and third quadrants, and arccos returns angles in the first and second quadrants, this guides selecting the correct angle measure for θ.

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