Questions: Determine a general formula (or formulas) for the solutions to the following equation. Then, determine the specific solutions (if any) on the interval [0,2 pi ). cos theta = 1/sqrt(2) Describe in the most concise way possible the general solution to the given equation, where k is any integer. Select the correct choice below and fill in all the answer boxes within your choice. (Simplify your answers. Type any angle measures in radians. Use the smallest non-negative angle possible when describing the general form of each angle. Use ascending order, entering the expression based on the smallest angle first. Type exact answers, using pi as needed. Use integers or fractions for any numbers in the expressions.) A. theta = + k B. theta = pi/3 + 2 pi k or theta = 5 pi/3 + 2 pi k C. theta = + k or theta = + k or theta = + k D. theta = + k or theta = + k or theta = k or theta = + k E. There is no solution.

Determine a general formula (or formulas) for the solutions to the following equation. Then, determine the specific solutions (if any) on the interval [0,2 pi ). cos theta = 1/sqrt(2) Describe in the most concise way possible the general solution to the given equation, where k is any integer. Select the correct choice below and fill in all the answer boxes within your choice. (Simplify your answers. Type any angle measures in radians. Use the smallest non-negative angle possible when describing the general form of each angle. Use ascending order, entering the expression based on the smallest angle first. Type exact answers, using pi as needed. Use integers or fractions for any numbers in the expressions.) A. theta = + k B. theta = pi/3 + 2 pi k or theta = 5 pi/3 + 2 pi k C. theta = + k or theta = + k or theta = + k D. theta = + k or theta = + k or theta = k or theta = + k E. There is no solution.
Transcript text: Determine a general formula (or formulas) for the solutions to the following equation. Then, determine the specific solutions (if any) on the interval $[0,2 \pi$ ). \[ \cos \theta=\frac{1}{\sqrt{2}} \] Describe in the most concise way possible the general solution to the given equation, where $k$ is any integer. Select the correct choice below and fill in all the answer boxes within your choice. (Simplify your answers. Type any angle measures in radians. Use the smallest non-negative angle possible when describing the general form of each angle. Use ascending order, entering the expression based on the smallest angle first. Type exact answers, using $\pi$ as needed. Use integers or fractions for any numbers in the expressions.) A. $\theta=$ $\square$ $+$ $\square$ k B. $\theta=\frac{\pi}{3}+2 \pi k$ or $\theta=\frac{5 \pi}{3}+2 \pi k$ C. $\theta=\square+\square \mathrm{k}$ or $\theta=\square+\square \mathrm{k}$ or $\theta=\square+\square \mathrm{k}$ D. $\theta=$ $\square$ $+$ $\square$ k or $\theta=\square+\square \mathrm{k}$ or $\theta=$ $\square$ k or $\theta=$ $\square$ $+$ $\square$ k E. There is no solution. 1
failed

Solution

failed
failed

Solution Steps

To solve the equation \(\cos \theta = \frac{1}{\sqrt{2}}\), we recognize that this corresponds to the angles where the cosine value is \(\frac{1}{\sqrt{2}}\), which are \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{7\pi}{4}\) within the interval \([0, 2\pi]\). The general solution for \(\cos \theta = \frac{1}{\sqrt{2}}\) is given by \(\theta = \frac{\pi}{4} + 2\pi k\) or \(\theta = \frac{7\pi}{4} + 2\pi k\), where \(k\) is any integer. These solutions account for the periodic nature of the cosine function.

Step 1: Identify the General Solution

The equation \(\cos \theta = \frac{1}{\sqrt{2}}\) has general solutions given by: \[ \theta = \frac{\pi}{4} + 2\pi k \quad \text{and} \quad \theta = \frac{7\pi}{4} + 2\pi k \] where \(k\) is any integer.

Step 2: Determine Specific Solutions in the Interval \([0, 2\pi]\)

To find specific solutions within the interval \([0, 2\pi]\), we evaluate the general solutions for \(k = 0\):

  • For \(k = 0\): \[ \theta_1 = \frac{\pi}{4} \approx 0.7854 \] \[ \theta_2 = \frac{7\pi}{4} \approx 5.4978 \]

Both values fall within the interval \([0, 2\pi]\).

Step 3: List the Specific Solutions

The specific solutions to the equation \(\cos \theta = \frac{1}{\sqrt{2}}\) in the interval \([0, 2\pi]\) are: \[ \theta_1 = \frac{\pi}{4} \quad \text{and} \quad \theta_2 = \frac{7\pi}{4} \]

Final Answer

The specific solutions are: \[ \boxed{\frac{\pi}{4}, \frac{7\pi}{4}} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
The figure to the right shows the average price of silver for recent years. Let S(t) represent the average price (in dollars per ounce) of silver for year t. Answer parts (a) through ( g ) below. (a) Is S(t) a linear function? A. No, because it increases and decreases B. Yes, because it increases C. Yes, because it consists of a collection of straight line segments D. No because S(t) is not a multiple of t (b) What is the independent variable? A. ounces, x B. year, t C. average price of silver, S(t) D. dollars, S (c) What is the dependent variable? A. average price of silver, S(t) B. year, t C. dollars, x D. ounces, S (d) What is the domain of the function? The domain is . (Type your answer in interval notation.)
failedAnswered
a. secrete hormones into the bloodstream, whereas secrete substances into ducts and onto the skin or the lumen of a hollow organ. b. Goblet cells and mammary glands are both exocrine glands-how are they similar and how are they different?
failedAnswered
Add. 23.4+7.88+0.9848 23.4+7.88+0.9848=
failedAnswered
In the 50-30-20 Rule, 50% of your income should be spent on
failedAnswered
Chapter 15: "What Is Freedom?": Reconstruction, 1865-1877 15.3. What were the social and political effects of Radical Reconstruction in the South? Match the descriptions of white Republicans in the South during Reconstruction to the correct term. These northern whites relocated to the South after the Civil War. Southern whites believed they had come simply for economic gain. These southern whites supported the Republicans and had not supported secession or the Confederacy. These were wartime Unionists from the South who hoped Reconstruction governments would help them recover from wartime economic losses. Some were investors in land and railroads who saw the postwar South as an opportunity to combine personal financial gain with a role in helping to substitute "the civilization of freedom for that of slavery."