Problem 21 - Algebra 2

Question

Solve each equation in the interval from 0 to 2\(\pi .\) Round to the nearest hundredth. \(8 \cos \frac{\pi}{3} t=5\)

Step-by-Step Solution

Answer

The two solutions to the equation in the range from 0 to \(2\pi\) are approximately \(t=1.34\) and \(t=5.09\), rounded to the nearest hundredths.

Step 1: Isolate the cosine function

To isolate the cosine function, we divide both sides of our initial equation by 8: \[\frac{8 \cos \frac{\pi}{3} t}{8}=\frac{5}{8}\]Which simplifies to: \[\cos \frac{\pi}{3} t=\frac{5}{8}\]

Step 2: Apply the inverse cosine function

Next, we apply the inverse cosine function or \(\arccos\) (also written as \(\cos^{-1}\)) to both sides to solve for \(t\): \[\frac{\pi}{3} t=\arccos\left(\frac{5}{8}\right)\] We calculate \(\arccos\left(\frac{5}{8}\right)\) to get an approximate result:

Step 3: Solve for t

Now we can solve for \(t\) by dividing both sides by \(\frac{\pi}{3}\): \[t=\frac{3}{\pi}\arccos\left(\frac{5}{8}\right)\] To find the value for \(t\) we compute the right hand side of this equation, taking care of the rounding to the nearest hundredth as instructed in the problem. Additionally, since the cosine function has a periodicity of \(2\pi\) (meaning it repeats every \(2\pi\) interval), we should consider the solutions in the range from 0 to \(2\pi\). Therefore, if \(t = \frac{3}{\pi}\arccos\left(\frac{5}{8}\right)\) is a solution, then so is \(2\pi - t\)

Step 4: Find all solutions

Therefore, the two solutions in the interval from 0 to \(2\pi\) are: \(t = \frac{3}{\pi}\arccos\left(\frac{5}{8}\right)\) and \(t = 2\pi - \frac{3}{\pi}\arccos\left(\frac{5}{8}\right)\). We calculate these values to obtain the final response.

Key Concepts

Cosine Function

The cosine function is fundamental in trigonometry and represents one of the basic trigonometric functions. It's often abbreviated as "cos." The essence of the cosine function is its periodic nature, forming a wave pattern that oscillates between -1 and 1.

Key characteristics of the cosine function are:

• It is periodic, meaning it repeats its values at regular intervals known as the period, which is typically every \(2\pi\) radians for the cosine function.
• It is an even function, meaning \(\cos(-x) = \cos(x)\).
• The cosine of 0 degrees (or 0 radians) is 1, while the cosine of 90 degrees (or \(\frac{\pi}{2}\) radians) is 0.

For the equation \(8 \cos \frac{\pi}{3} t=5\), the approach involves isolating the cosine function. This action simplifies the problem to finding the angle where the cosine value equals \(\frac{5}{8}\). Understanding how to work with the cosine function is essential for solving trigonometric equations like this one.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when we know the values of the trigonometric functions. These functions are crucial for solving trigonometric equations.

The cosine inverse, denoted as \(\arccos\) or \(\cos^{-1}\), helps us determine the angle whose cosine is a given number.

• \(\arccos(x)\) gives the angle \(\theta\) such that \(\cos(\theta) = x\).
• The range of \(\arccos\) is from 0 to \(\pi\) radians, representing all possible angles for the inverse cosine.

In the problem \(8 \cos \frac{\pi}{3} t=5\), applying \(\arccos\) to both sides transforms the equation into \(\frac{\pi}{3} t = \arccos\left(\frac{5}{8}\right)\). This step is critical as it changes the focus from finding the cosine value to finding the angle itself (i.e., \(t\)).

Given the periodic nature of cosine, there might be multiple solutions in any given range, which is why calculating both \(t\) and \(2\pi-t\) is necessary.

Radian Measure

The radian measure is a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians divide a circle into \(2\pi\) parts, which makes them more natural for use in trigonometry and calculus.

Understanding radians is straightforward if you keep in mind these key points:

• One complete revolution around a circle (360 degrees) is equivalent to \(2\pi\) radians.
• Thus, \(180\) degrees equals \(\pi\) radians, making it easier to convert between these two measurement systems.
• Radians provide a direct correlation between the angle and the length of the arc that subtends it.

In the provided exercise, radian measure is employed when considering the solutions \(t = \frac{3}{\pi} \arccos\left(\frac{5}{8}\right)\). By working in radians, calculations involving circular arcs and trigonometric identities are greatly simplified. Moreover, representing angles in the interval from 0 to \(2\pi\) accentuates the periodic properties of trigonometric functions.