Problem 28 - Algebra and Trigonometry

Question

\(25-38\) . Find all solutions of the given equation. $$ \sqrt{2} \cos \theta-1=0 $$

Step-by-Step Solution

Answer

The solutions are \( \theta = \frac{\pi}{4} + 2n\pi \) or \( \theta = \frac{7\pi}{4} + 2n\pi \), where \( n \) is an integer.

Step 1: Understand the Equation

The equation given is \( \sqrt{2} \cos \theta - 1 = 0 \). We need to find the values of \( \theta \) that make this equation true.

Step 2: Solve for \( \cos \theta \)

Add 1 to both sides of the equation to isolate the \( \cos \theta \) term: \( \sqrt{2} \cos \theta = 1 \).

Step 3: Divide by \( \sqrt{2} \)

Divide both sides of the equation by \( \sqrt{2} \) to solve for \( \cos \theta \): \( \cos \theta = \frac{1}{\sqrt{2}} \).

Step 4: Simplify the Right Side

Recognize that \( \frac{1}{\sqrt{2}} \) simplifies to \( \frac{\sqrt{2}}{2} \), so \( \cos \theta = \frac{\sqrt{2}}{2} \).

Step 5: Determine Possible Angles

Recall that \( \cos \theta = \frac{\sqrt{2}}{2} \) corresponds to angles of \( \theta = \frac{\pi}{4} + 2n\pi \) or \( \theta = \frac{7\pi}{4} + 2n\pi \), where \( n \) is any integer. This is because cosine is positive in the first and fourth quadrants.

Key Concepts

Cosine Function

The cosine function, often denoted as \( \cos \theta \), is one of the primary trigonometric functions. It associates an angle with a value, which represents the ratio of the adjacent side to the hypotenuse in a right triangle. The range of the cosine function is from -1 to 1 due to the nature of these geometric relationships.
The function is periodic, which means it repeats its values in regular intervals. Specifically, its period is \( 2\pi \). This is crucial for solving trigonometric equations as it defines the repeated nature of the solutions.
In trigonometry, understanding these concepts allows you to manipulate and solve equations involving cosine, like the one given in the exercise. By isolating \( \cos \theta \) and understanding its periodicity, you can find multiple solutions for any given angle.

Angle Solutions

When dealing with trigonometric equations like \( \cos \theta = \frac{\sqrt{2}}{2} \), finding angle solutions involves understanding where this condition holds true on the unit circle. The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane.
The particular cosine value \( \frac{\sqrt{2}}{2} \) specifically refers to certain key angles. In standard position, these angles are \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{7\pi}{4} \).

• \( \frac{\pi}{4} \) is found in the first quadrant.
• \( \frac{7\pi}{4} \) is located in the fourth quadrant.

These angles are found by comparing the cosine value to known reference angles in trigonometry, allowing you to determine the general solutions.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each representing a distinct area where different trigonometric functions have positive or negative values. Knowing the quadrants is important for solving trigonometric equations as it helps to determine where cosine is positive.

• **First Quadrant (0 to \( \frac{\pi}{2} \))**: Both sine and cosine are positive here.
• **Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \))**: Sine is positive, cosine is negative.
• **Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \))**: Both sine and cosine are negative.
• **Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \))**: Sine is negative, cosine is positive.

Given the equation \( \cos \theta = \frac{\sqrt{2}}{2} \), the solutions \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{7\pi}{4} \) lie in the first and fourth quadrants, respectively. Thus, understanding the properties of these quadrants helps to accurately locate the correct angle solutions.