In Exercises 1–26, find the exact value of each expression._cos⁻¹ (- √2/2)

Verified step by step guidance

1. Recognize that \( \cos^{-1} \) is the inverse cosine function, which gives the angle whose cosine is the given value.
2. Identify the range of the \( \cos^{-1} \) function, which is \([0, \pi]\) or \([0, 180^\circ]\).
3. Recall that \( \cos(\theta) = -\frac{\sqrt{2}}{2} \) corresponds to angles in the second quadrant, where cosine is negative.
4. Determine the reference angle where \( \cos(\theta) = \frac{\sqrt{2}}{2} \), which is \( \frac{\pi}{4} \) or \( 45^\circ \).
5. Find the angle in the second quadrant by subtracting the reference angle from \( \pi \), resulting in \( \pi - \frac{\pi}{4} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹ (arccosine), are used to find the angle whose cosine is a given value. For example, cos⁻¹(x) returns the angle θ such that cos(θ) = x. Understanding how to interpret these functions is crucial for solving problems involving angles and their corresponding trigonometric ratios.

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the relationship between angles and their sine and cosine values. It is a circle with a radius of one centered at the origin of a coordinate plane. The coordinates of points on the unit circle correspond to the cosine and sine of the angle formed with the positive x-axis, which is essential for determining the exact values of trigonometric functions.

Quadrants and Angle Values

In trigonometry, the coordinate plane is divided into four quadrants, each affecting the signs of the sine and cosine values. For cos⁻¹(-√2/2), it is important to recognize that the cosine value is negative, which occurs in the second and third quadrants. Knowing the specific angles that correspond to these cosine values helps in accurately determining the angle from the inverse function.