Problem 54
Question
The graph of the equation \(y=2 \cos \theta\) is shown. Which is a solution for \(2 \cos \theta=1 ?\) F. \(\frac{8 \pi}{3}\) G. \(\frac{13 \pi}{3}\) H. \(\frac{10 \pi}{3}\) J. \(\frac{15 \pi}{3}\)
Step-by-Step Solution
Verified Answer
\(\theta = \frac{13\pi}{3}\) (Option G) is a solution.
1Step 1: Set Up the Equation
The first task is to verify the equation given in the problem: \[2 \cos \theta = 1\] We need a value of \(\theta\) such that plugging \(\theta\) into the equation yields a true statement.
2Step 2: Solve for \(\theta\)
Divide both sides of the equation by 2 to isolate \(\cos \theta\):\[\cos \theta = \frac{1}{2}\]Determine the values of \(\theta\) that satisfy this equation.
3Step 3: Identify Solutions on the Unit Circle
On the unit circle, \(\cos \theta = \frac{1}{2}\) at \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{5\pi}{3}\). These angles correspond to the positive \(x\)-coordinates on the circle where the cosine value is \(\frac{1}{2}\).
4Step 4: Consider General Solutions
The cosine function has a period of \(2\pi\), so general solutions are given by:\[\theta = \frac{\pi}{3} + 2k\pi\, \text{and}\, \theta = \frac{5\pi}{3} + 2k\pi\]where \(k\) is any integer.
5Step 5: Calculate Specific Solutions
Substitute various integers into the general solution to find which one matches the options:- For \(\theta = \frac{\pi}{3} + 2(1)\pi = \frac{7\pi}{3}\)- For \(\theta = \frac{5\pi}{3} + 2(1)\pi = \frac{11\pi}{3}\)- For \(\theta = \frac{\pi}{3} + 2(2)\pi = \frac{13\pi}{3}\)- For \(\theta = \frac{5\pi}{3} + 2(2)\pi = \frac{17\pi}{3}\)
6Step 6: Match with Given Choices
Among the calculated solutions, \(\theta = \frac{13\pi}{3}\) matches one of the choices provided (option G). Therefore, \(\theta = \frac{13\pi}{3}\) is a solution for the given equation.
Key Concepts
Cosine FunctionUnit CircleGeneral SolutionsPeriodic Functions
Cosine Function
The cosine function is one of the fundamental trigonometric functions, abbreviated as \(\cos\). It helps to relate the angles and sides of a right triangle. This function takes an angle as an input and returns a ratio known as the cosine of the angle. Specifically, for an angle in a right-angled triangle, it describes the ratio of the length of the adjacent side to the hypotenuse.
Cosine is periodic, repeating its values in a regular pattern. It is particularly useful in describing oscillations, waves, and circles.
Cosine is periodic, repeating its values in a regular pattern. It is particularly useful in describing oscillations, waves, and circles.
- The cosine of 0° is 1.
- Cosine reaches 0 at 90° and 270°.
- Completes a full cycle every 360° or \(2\pi\) radians.
Unit Circle
The unit circle is a circle with a radius of one, centered at the origin of a coordinate system. In trigonometry, it's a powerful tool for understanding angles and how trigonometric functions like cosine work.
On the unit circle, each point's coordinates represent the cosine and sine of the angle formed by the point, the origin, and the positive x-axis. Specifically, the x-coordinate of a point on the unit circle is the cosine of the angle, while the y-coordinate is the sine.
On the unit circle, each point's coordinates represent the cosine and sine of the angle formed by the point, the origin, and the positive x-axis. Specifically, the x-coordinate of a point on the unit circle is the cosine of the angle, while the y-coordinate is the sine.
- Provides a simple representation for angles in terms of radians.
- Helps visualize how values repeat for angles greater than \(360°\) or \(2\pi\).
- Allows identification of cosine values at particular angles, such as \(\frac{\pi}{3}\) and \(\frac{5\pi}{3}\) where \(\cos \theta = \frac{1}{2}\).
General Solutions
In trigonometric equations, finding general solutions means finding all possible angles, \(\theta\), that solve the equation. Because trigonometric functions like cosine are periodic, there are often infinitely many solutions that form a pattern.
For an equation like \(\cos \theta = \frac{1}{2}\), the general solutions account for the repeating nature of the cosine function:
For an equation like \(\cos \theta = \frac{1}{2}\), the general solutions account for the repeating nature of the cosine function:
- Expressed with formulas that include the base solution \(\theta_0\) and multiples of the function's period.
- Example: \(\theta = \frac{\pi}{3} + 2k\pi\) and \(\theta = \frac{5\pi}{3} + 2k\pi\), where \(k\) is any integer.
Periodic Functions
Periodic functions repeat their values after a certain interval, known as the period. The cosine function is a perfect example of a periodic function, with a period of \(2\pi\) or 360°, meaning its values repeat every \(2\pi\).
The periodicity of the cosine function allows for multiple solutions to trigonometric equations like \(2\cos \theta = 1\). Here’s why periodicity matters:
The periodicity of the cosine function allows for multiple solutions to trigonometric equations like \(2\cos \theta = 1\). Here’s why periodicity matters:
- It helps predict the values of the function at any given angle based on past cycles.
- Essential for solving equations and understanding oscillating behaviors in physics and engineering.
- The angles lead to solutions that repeat at regular intervals defined by the period.
Other exercises in this chapter
Problem 53
Find each value. $$ \sin \left(\cos ^{-1} \frac{2}{3}\right) $$
View solution Problem 53
Find the first five terms of the sequence in which \(a_{1}=3, a_{n+1}=2 a_{n}+5\)
View solution Problem 54
Simplify each expression. $$ \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} $$
View solution Problem 54
Simplify each expression. \(\csc \theta \tan \theta+\sec \theta\)
View solution