Problem 45 - Precalculus: Functions and Graphs

Question

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 \(x = \pi/4, 7\pi/4, 9\pi/4, 15\pi/4\).

Step 1: Identify the Key Angles

Recall that cosine of an angle equals \(\sqrt{2}/2\) at angles \(\pi/4\) and \(7\pi/4\) within one full circle \([0, 2\pi]\).

Step 2: Extend to the Given Interval

Since we are considering the interval \([0, 4\pi]\), extend the key angles by adding multiples of \(2\pi\) within this interval: - \(\pi/4 + 2\pi = 9\pi/4\) - \(7\pi/4 + 2\pi = 15\pi/4\)

Step 3: Verify the Solutions

Confirm that each angle is within the specified interval \([0, 4\pi]\). Every calculated \(x\) must satisfy the equation \(\cos x = \sqrt{2} / 2\), and the angles \(\pi/4, 7\pi/4, 9\pi/4, 15\pi/4\) all meet these conditions. The value of \(\cos\) at these angles is indeed \(\sqrt{2}/2\).

Key Concepts

cosine function

The cosine function, often represented as \( \cos x \), is a fundamental trigonometric function. It relates an angle \( x \) in a right triangle to the ratio of the adjacent side over the hypotenuse. The cosine function is periodic, meaning it repeats its values in regular intervals. This periodicity occurs every \( 2\pi \) radians, equivalent to 360 degrees.

• Periodic Nature: The cosine curve is symmetrical and repetitive, making it handy for modeling cycles like waves and oscillations.
• Range: The values of \( \cos x \) only range between \(-1\) and \(1\), covering all possible values the function can take.
• Key Points: The function hits its maximum at \(\cos 0 = 1\) and minimum at \(\cos \pi = -1\).

For this exercise, you need to find angles \( x \) such that \( \cos x = \sqrt{2}/2 \). Knowing this value appears at key angles simplifies locating \( x \) on the unit circle and extending it to larger intervals.

unit circle

The unit circle is a circle with a radius of 1, typically centered at the origin of a coordinate plane. This simple tool is incredibly powerful for solving trigonometric equations as it vividly presents sine and cosine values as coordinates.

• Coordinates: The horizontal axis represents cosine values and the vertical axis represents sine values.
• Radians: Positions on the unit circle are expressed in radians, providing a continuous measurement of angles.
• Symmetry: Because of its symmetry, the unit circle can show repeating patterns of sine and cosine values, helping to find solutions over larger intervals.

For \( \cos x = \sqrt{2}/2\), you find the points on the unit circle where the x-coordinate (or cosine value) matches that ratio. This happens at angles \( \pi/4 \) and \( 7\pi/4 \) initially, as seen directly on the unit circle's axis split by these ratios.

angle measurement

Angles can be measured in degrees or radians. In trigonometry, radians are often more convenient, especially when dealing with periodic functions like sine and cosine.

• Conversion: There are \(2\pi\) radians in a full circle, equivalent to 360 degrees, so \(1\pi\) radian equals 180 degrees.
• Using Radians: Radians allow for precise calculation of angles and are integral when using mathematical formulas.
• Standard and Extended Ranges: Equations often solve within the basic interval \([0, 2\pi]\), but extending these solutions to intervals like \([0, 4\pi]\) involves adding multiples of \(2\pi\).

In this problem, once we identify the initial angles, we calculate further solutions by adding \(2\pi\) to them, thus expanding the solution set in the extended interval. This approach leverages the inherent periodicity of trigonometric functions, creating comprehensive solution sets for given conditions.