Problem 102 - Algebra and Trigonometry

Question

Find the exact value of each expression. Do not use a calculator. $$ \cos \left[\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)-\sin ^{-1}\left(-\frac{1}{2}\right)\right] $$

Step-by-Step Solution

Answer

The exact value of the given expression is -1

Step 1: Determine the values of the inverse trigonometric functions

Find the values of \(\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)\) and \(\sin^{-1}\left(-\frac{1}{2}\right)\). Using the unit circle, we know that \(\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) = \frac{5\pi}{6}\), and \(\sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6}\)

Step 2: Substitute the values in the expression

Replace \(\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)\) and \(\sin^{-1}\left(-\frac{1}{2}\right)\) with \(\frac{5\pi}{6}\) and \(-\frac{\pi}{6}\) in the expression respectively. The expression becomes \(\cos(\frac{5\pi}{6} - (-\frac{\pi}{6})) = \cos(\frac{5\pi}{6} + \frac{\pi}{6})\)

Step 3: Simplify the expression within the cosine

Combine \(\frac{5\pi}{6}\) and \(\frac{\pi}{6}\) to get \(\cos(\pi)\)

Step 4: Determine the value of cosine

From the unit circle we know that \(\cos(\pi) = -1\)

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that deeply influences how we understand angles and trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate system. The unit circle allows us to connect angles with coordinates on the circle and is essential in defining the sine, cosine, and tangent of angles.

When working with the unit circle, any angle can be represented as a point \( (x, y) \), where \( x = \cos(\theta) \) and \( y = \sin(\theta) \). Here's why the unit circle is so useful:

• **Angle Representation:** Each angle, measured from the positive x-axis, corresponds to a specific point on the circle.
• **Coordinates Connection:** The coordinates (x, y) of points on the circle easily give us values for the cosine and sine functions, respectively.
• **Inverse Trigonometric Functions:** Inverse functions like \( \cos^{-1} \) and \( \sin^{-1} \) use the unit circle to determine angles for specific cosine and sine values.

Understanding the unit circle helps simplify problems involving inverse trigonometric functions, as it allows you to visualize and derive angles simply.

Cosine Function

The cosine function is one of the primary trigonometric functions and is key to understanding relationships between angles and sides in right triangles. On the unit circle, the cosine of an angle \( \theta \) is the x-coordinate of the point where the terminal side of the angle intersects the circle. This intersection insight helps facilitate inverse calculations.

Here are some important properties of the cosine function to remember:

• **Periodicity:** The cosine function repeats every \( 2\pi \) radians, meaning \( \cos(\theta + 2\pi) = \cos(\theta) \).
• **Symmetry:** The function is even, which implies \( \cos(-\theta) = \cos(\theta) \).
• **Inverse Relationship:** Inverse cosine, \( \cos^{-1}(x) \), gives the angle whose cosine is \( x \), with a principal range between 0 and \( \pi \).

When you understand cosine's geometrical representation on the unit circle, solving representations like \( \cos(\pi) = -1 \), as in the exercise provided, becomes straightforward.

Trigonometric Identities

Trigonometric identities are equations that hold true for all angles, central to simplifying and solving trigonometric problems. They express relationships between the trigonometric functions and are essential tools in any angle or function transformation.

Here are some of the key identities to know:

• **Pythagorean Identity:** \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• **Sum and Difference Identities:** Used when you add or subtract angles. For cosine, it states that \( \cos(a \pm b) = \cos(a) \cos(b) \mp \sin(a) \sin(b) \).
• **Double Angle Identity:** Such as \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \).

These identities were used implicitly in the original problem solution, where the focus was on simplifying expressions involving angle sums within trigonometric functions. Understanding these identities allows you to maneuver through complex expressions effortlessly.