Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.


cos⁻¹ (- 1/2 )

Verified step by step guidance

1. Understand that ${\cos }^{-1}(x)$ represents the angle whose cosine is $x$.
2. Recall that the range of ${\cos }^{-1}(x)$ is $[0,\pi ]$.
3. Identify the angle $\theta$ in the range $[0,\pi ]$ such that $\cos (\theta )=-\frac{1}{2}$.
4. Recognize that $\cos (\pi -\theta )=-\cos (\theta )$, and since $\cos (\frac{\pi }{3})=\frac{1}{2}$, then $\cos (\pi -\frac{\pi }{3})=-\frac{1}{2}$.
5. Conclude that $\theta =\pi -\frac{\pi }{3}$ is the angle in the range $[0,\pi ]$ such that $\cos (\theta )=-\frac{1}{2}$.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsine and arccosine, are used to find angles when given a ratio of sides in a right triangle. For example, cos⁻¹(x) gives the angle whose cosine is x. These functions are essential for solving problems where the angle is unknown, and they have specific ranges to ensure each output is unique.

Range of Inverse Cosine

The range of the inverse cosine function, cos⁻¹(x), is restricted to the interval [0, π]. This means that when evaluating cos⁻¹(-1/2), the resulting angle must fall within this range. Understanding this range is crucial for correctly interpreting the output of the inverse cosine function.

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps visualize the values of trigonometric functions. It is a circle with a radius of one centered at the origin of a coordinate plane. By using the unit circle, one can determine the angles corresponding to specific cosine values, such as -1/2, which corresponds to angles in the second and third quadrants.