In Exercises 1–26, find the exact value of each expression._cos⁻¹ √3/2

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 given value: \( \frac{\sqrt{3}}{2} \).
3. Recall that \( \cos(\theta) = \frac{\sqrt{3}}{2} \) corresponds to specific angles on the unit circle.
4. Determine the angle \( \theta \) in the range \([0, \pi]\) (since \( \cos^{-1} \) returns values in this range) that satisfies \( \cos(\theta) = \frac{\sqrt{3}}{2} \).
5. Conclude that the angle is one of the standard angles, which you can find by recalling the unit circle values.

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(√3/2) asks for the angle θ where cos(θ) = √3/2. The range of the arccosine function is from 0 to π radians, which is essential for determining the correct angle.

Unit Circle

The unit circle is a fundamental concept in trigonometry that defines the relationship between angles and their corresponding sine and cosine values. It is a circle with a radius of one centered at the origin of a coordinate plane. Understanding the unit circle helps in identifying the angles that yield specific cosine values, such as √3/2, which corresponds to angles of π/6 and 11π/6.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to specific angles where the sine, cosine, and tangent values can be expressed as simple fractions or radicals. For instance, cos(π/6) = √3/2 is an exact value. Knowing these exact values allows for quick calculations and helps in solving problems without relying on calculators.