darshanpatel said:Homework Statement
Find all the solutions of each equation in the interval (0, 2∏)
cos(2x)=1/2
Homework Equations
-None-
The Attempt at a Solution
cos(2x)=1/2
2x=(cos^(-1))*1/2
2x= ((∏)/3), ((5∏)/3)
2x= ((∏)/3), ((5∏)/3), ((14∏)/6)
x= ∏/6, ((5∏)/6),
That's all I have.
I know there is one more solution, but I don't know how to find it.
Please help, and thanks in advance.
darshanpatel said:What do you mean by corresponding range of 2x? I don't understand...
To solve an equation like cos(2x) = 1/2 in the interval (0, 2∏), you can use the double angle formula for cosine: cos(2x) = cos²(x) - sin²(x). Rewrite the equation as cos²(x) - sin²(x) = 1/2. Then, use the Pythagorean identity sin²(x) + cos²(x) = 1 to simplify the equation to 2cos²(x) - 1 = 1/2. Solving for cos(x), we get cos(x) = ± √(3)/2. Since we are looking for solutions in the interval (0, 2∏), we need to consider both the positive and negative solutions, which are cos(x) = √(3)/2 and cos(x) = -√(3)/2. This gives us two solutions for x: x = π/6 and x = 5π/6.
Yes, the unit circle can be a helpful tool in solving equations involving cos(2x). By plotting the cosine values on the unit circle, you can easily find the reference angles and use them to find the solutions in the given interval. In the example of cos(2x) = 1/2, the reference angle is π/6, which corresponds to the solutions x = π/6 and x = 5π/6.
Yes, there are other methods you can use to solve these types of equations. You can also use the half angle formula for cosine or the inverse cosine function. Additionally, you can graph both sides of the equation and find the intersection points, which represent the solutions.
If cos(2x) is equal to a negative value, such as -1/2 or -√(3)/2, you can follow the same steps as in question 1 to solve for x. However, keep in mind that the solutions you find will be in the form of an angle, so you may need to use inverse trigonometric functions to find the actual values for x.
Yes, you can use similar methods to solve equations involving other trigonometric functions such as sine and tangent. However, the identities and formulas used will be different. For example, to solve an equation like sin(2x) = 1/2, you would use the double angle formula for sine: sin(2x) = 2sin(x)cos(x). Then, use the Pythagorean identity and solve for sin(x), just like in question 1.