Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-46: Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$ \cos x=\sqrt{2} / 2 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{4}, \frac{7\pi}{4}, \frac{9\pi}{4}, \frac{15\pi}{4}\).

Step 1: Understanding the Equation

The equation given is \( \cos x = \frac{\sqrt{2}}{2} \). In trigonometry, we know that \( \cos(x) \) equals \( \frac{\sqrt{2}}{2} \) at specific angles on the unit circle. These angles are \( x = \frac{\pi}{4} \) and \( x = \frac{7\pi}{4} \) within the interval \([0, 2\pi]\). We must find all solutions in the interval \([0, 4\pi]\).

Step 2: Finding Solutions in [0, 2π]

Within the interval \([0, 2\pi]\), the solutions where \( \cos x = \frac{\sqrt{2}}{2} \) are \( x = \frac{\pi}{4} \) and \( x = \frac{7\pi}{4} \). These are primary solutions based on the symmetry and periodicity of the cosine function.

Step 3: Extending to [2π, 4π]

Cosine function is periodic with a period \(2\pi\). To find solutions in \([2\pi, 4\pi]\), add \(2\pi\) to each of the solutions from the first period: \( x = \frac{\pi}{4} + 2\pi \, \text{and} \, x = \frac{7\pi}{4} + 2\pi \). This results in \( x = \frac{9\pi}{4} \) and \( x = \frac{15\pi}{4} \).

Step 4: Compiling All Solutions

Simple from previous steps, we can compile all solutions of \( \cos x = \frac{\sqrt{2}}{2} \) in the interval \([0, 4\pi]\). These are: \( x = \frac{\pi}{4}, \frac{7\pi}{4}, \frac{9\pi}{4}, \text{and} \frac{15\pi}{4} \).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that arises in various geometric and practical contexts. It relates the measure of an angle in a right triangle to the ratios of lengths of the sides. Specifically, in a right triangle with angle \(x\), the cosine of \(x\) is the ratio of the adjacent side to the hypotenuse. This relationship is often expressed as:

• \( \cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \)

The cosine function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians. The graph of \(y = \cos(x)\) oscillates between \(-1\) and \(1\) and has key points where it takes on values of \(\pm1\) and 0. Finding where the cosine function equals a particular value, such as \(\frac{\sqrt{2}}{2}\), involves identifying angles on the unit circle where this ratio holds.
Because of its periodic nature, the cosine function will periodically return to the same value at regular intervals, which is fundamental when solving trigonometric equations over specified ranges.

Unit Circle

The unit circle is a powerful tool in trigonometry. Centered at the origin of the cartesian coordinate plane, it has a radius of 1. Each point \((x, y)\) on the circle corresponds to an angle \(x\), measured in radians or degrees, emanating from the positive x-axis.

• The cosine of an angle in the unit circle is the x-coordinate of the corresponding point.
• The sine of the angle is the y-coordinate of that point.

The unit circle enables a deeper understanding of trigonometric functions beyond right triangles, extending their definitions to encompass all real numbers. It aids in visualizing how the values of these functions depend on the angle as it "travels" around the circle.
For solving \( \cos(x) = \frac{\sqrt{2}}{2} \), the unit circle helps identify angles \( \frac{\pi}{4} \) and \( \frac{7\pi}{4} \), as those angles yield x-coordinates of \( \frac{\sqrt{2}}{2} \). The journey around the circle for full rotations helps extend these observations to multiple cycles, relevant in problems spanning larger intervals like \([0, 4\pi]\).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the occurring variables. They serve as essential tools for simplifying and solving trigonometric equations. Some critical identities include:

• Pythagorean Identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle Sum and Difference Identities: \( \cos(x \pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y) \)
• Double Angle Identities: \( \cos(2x) = \cos^2(x) - \sin^2(x) \)

These identities allow you to transform complex expressions into simpler forms, making them more manageable to work with. For example, understanding that the cosine function repeats every \(2\pi\) (the periodic identity) is crucial in extending solutions beyond a single cycle on the unit circle.
In solving \( \cos(x) = \frac{\sqrt{2}}{2} \) over larger intervals, recognizing these identities postulates consistency in outputs across multiples of oscillations, helping find all associated angles efficiently.