Problem 8 - Precalculus Essentials

Question

Find the exact value of each expression. $$\cos ^{-1} \frac{\sqrt{2}}{2}$$

Step-by-Step Solution

Answer

So the exact value of the given expression \(\cos^{-1}(\sqrt{2}/2)\) is \( \pi/4 \) radians or 45°.

Step 1: Recognize the given value in a Unit Circle

We know that in the Unit Circle, the cosine of 45° or \(\pi/4\) radians is \(\sqrt{2}/2\).

Step 2: Apply inverse cosine function to the value

By applying \(\cos^{-1}(x)\), we are looking for the angle whose cosine value is \(x\). From step 1, we can conclude that \(\cos^{-1}(\sqrt{2}/2)\) returns the angle which has cosine as \(\sqrt{2}/2\).

Key Concepts

Unit Circle

A **Unit Circle** is a fundamental tool in trigonometry that helps us understand and visualize the relationships between angles and trigonometric functions. This circle has a radius of 1 and is centered at the origin of a coordinate plane. When dealing with trigonometric functions, the unit circle makes it easy to retrieve values of sine, cosine, and tangent for common angles.

• The unit circle is divided into quadrants, each representing different angle measures.
• Coordinates on the unit circle can be represented as \( (\cos(\theta), \sin(\theta)) \).
• Angles measured counterclockwise from the positive x-axis are positive, while those measured clockwise are negative.

In context with the problem, the unit circle reveals that the angle whose cosine is \(\sqrt{2}/2\) is \(45°\) or \(\pi/4\) radians. That's because this angle in the first quadrant has such coordinates.

Cosine Function

The **Cosine Function** is one of the primary trigonometric functions and is crucial in understanding angle-side relationships in right-angled triangles, as well as circular movements. It relates the adjacent side to the hypotenuse in a right triangle but also has a broader application in the unit circle.

• The cosine of an angle \(\theta\) is the x-coordinate of the corresponding point on the unit circle.
• It is periodic and symmetric, repeating its values every \(2\pi\) radians.
• Cosine reaches its maximum value of 1 and minimum of -1.

In the case of the exercise given, by applying \(\cos^{-1} \), or arc cosine, to \(\sqrt{2}/2\), we identify the specific angle where cosine gives that value. In radians, this is \(\pi/4\), which is a fundamental concept depicted by the cosine function on the unit circle.

Radians

**Radians** are a way of measuring angles, essential for understanding trigonometry, calculus, and other mathematical areas. Unlike degrees, radians consider the radius of the circle and provide a natural method of angle measurement, especially useful in higher mathematics.

• There are \(2\pi \) radians in a full circle, equivalent to \(360°\).
• One radian is the angle subtended by an arc that is equal in length to the radius of the circle.
• Commonly used in trigonometry because they simplify many mathematical formulas.

For the exercise, the angle whose cosine value is \(\sqrt{2}/2\) is \(\pi/4\) radians. This equivalence is important and often used, as radians directly relate linear measurements with circular movements, making them instrumental in mathematical calculations. Learning to convert and understand these angles in radians is a key skill in trigonometry.