1

Problem

Determine the exact solutions for the equation \$ 2cos^2θ - 3cosθ + 1 = 0 "within the interval" 0 le θ < 2π \$.

2

Step

The given equation

\$ 2cos^2θ - 3cosθ + 1 = 0; 0 le θ < 2π \$

3

Formula

To solve the quadratic equation: \$cos\theta = (- b ± (\sqrt(b^2 - 4ac)))/ (2a) \$.

4

Hint

Here a = 2 , b = - 3 and c = 1.

5

Step

Now plug the values in the formula:

\$cos\theta = (- (- 3) ± (\sqrt((- 3)^2 - 4 \times 2 \times 1))) / (2 \times 2)\$

6

Step

After simplification:

\$cos\theta = (3 ± (\sqrt(9 - 8))) /4\$

\$cos\theta = (3 ± 1) / 4\$

7

Step

The equation yields two distinct solutions for the cosine function:

\$cos\theta = (3 + 1)/4\$ and \$cos\theta = (3 - 1)/4\$

\$cos\theta = 1\$ and \$cos\theta = 1/2\$

8

Step

Cosine equals 1 at 0 degrees and 360 degrees (or any multiple of 360 degrees):

\$\theta_1 = 0\$

\$\theta_2 = (2pi)\$

9

Step

When \$cos⁡(θ) = 1/2\$. This happens in the first and fourth quadrants. Using the unit circle or inverse cosine function, we find two angles:

\$\theta_3 = (pi)/3\$

\$\theta_4 = (5pi)/3\$

10

Solution

Therefore, the solutions to the equation \$2cos^⁡2(θ) - 3cos⁡(θ) + 1 = 0\$ on the interval 0 ≤ θ< 2π are \$θ = 0, (2pi), (pi)/3 and (5pi)/3\$ .

11

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

12

Choice-A

This option includes solutions \$pi\$ and \$pi/6\$, but it doesn’t include solution 0 or \$((5 pi)/3)\$, which are part of the correct solutions

Wrong \$ pi, pi/6, (7pi)/3 \$

13

Choice-B

This option includes \$pi/2\$ and \$pi/4\$, but it lacks the solution 0 or \$((5 pi)/3)\$, which are part of the correct solutions

Wrong \$ pi/2, pi/4, (3pi)/4 \$

14

Choice-C

This option includes \$pi/2\$, \$pi/4\$ and \$(3 pi)/4\$, but it doesn’t include solution 0 or \$((5 pi)/3)\$, which are part of the correct solutions

Wrong \$ pi/2, pi/4, (3pi)/4, (5pi)/4 \$

15

Choice-D

Option D accurately captures all the valid solutions for the equation within the specified range. The other options either miss some solutions or include irrelevant values

Correct \$ 0, 2pi, pi/3, (5pi)/3 \$

16

Answer

Option

D

17

Sumup

Please summarize choices