2 cosx - 1/cosx = 1
(2cos^2 x - 1)cosx = 1
2cos^2 x - 1 = cosx
2cos^2 x - cosx - 1 = 0
(2cosx + 1)(cosx-1) = 0
cosx = -1/2 or cosx = 1
x = 120º, 240º,0º,360º
2cosx-secx = 1
(2cos^2 x - 1)cosx = 1
2cos^2 x - 1 = cosx
2cos^2 x - cosx - 1 = 0
(2cosx + 1)(cosx-1) = 0
cosx = -1/2 or cosx = 1
x = 120º, 240º,0º,360º
The equation becomes:
2cosx - 1/cosx = 1
To eliminate the fraction, we can multiply both sides of the equation by cosx:
2cos^2x - 1 = cosx
Rearrange the equation:
2cos^2x - cosx - 1 = 0
Now we can solve this quadratic equation. Let's substitute cosx with a variable, such as t:
2t^2 - t - 1 = 0
To factor this quadratic equation, we set it equal to zero and factor:
(2t + 1)(t - 1) = 0
Setting each factor equal to zero and solving for t:
2t + 1 = 0 --> t = -1/2
t - 1 = 0 --> t = 1
Now substitute cosx back in for t:
cosx = -1/2 or cosx = 1
To find the values of x, we can take the inverse cosine of both sides:
x = arccos(-1/2) or x = arccos(1)
Using a calculator or trigonometric table, we find:
x = 120°, 240° or x = 0°, 360°
Therefore, the exact solutions for the equation 2cosx - secx = 1 on the interval [0, 360) are x = 0°, 120°, 240°, and 360°.
Start by rewriting sec(x) as 1/cos(x):
2cos(x) - 1/cos(x) = 1
To combine like terms, multiply both sides of the equation by cos(x):
2cos^2(x) - 1 = cos(x)
Now, rearrange the equation by moving all the terms to one side:
2cos^2(x) - cos(x) - 1 = 0
This is a quadratic equation in terms of cos(x). Let's solve for cos(x) by factoring:
(2cos(x) + 1)(cos(x) - 1) = 0
Setting each factor equal to zero gives us two separate equations:
2cos(x) + 1 = 0 ---> cos(x) = -1/2
cos(x) - 1 = 0 ---> cos(x) = 1
Now, we solve each equation for x.
For cos(x) = -1/2:
To find the values of x where cos(x) is equal to -1/2 on the interval [0, 360), we can refer to the unit circle. The unit circle is a circle with a radius of 1, centered at the origin (0,0), with angles measured counterclockwise from the positive x-axis.
On the unit circle, the values of x where cos(x) is equal to -1/2 are 120° and 240°.
For cos(x) = 1:
The only value of x on the interval [0, 360) where cos(x) is equal to 1 is 0°.
Therefore, the exact solutions to the equation 2cos(x) - sec(x) = 1 on the interval [0, 360) are x = 0°, 120°, and 240°.