Find the exact solutions to each equation for the interval [0,360)

2cosx-secx = 1

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

To find the exact solutions for the equation 2cosx - secx = 1 on the interval [0, 360), we can simplify it by replacing secx with its reciprocal, which is 1/cosx.

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°.

To find the exact solutions to the equation 2cos(x) - sec(x) = 1 on the interval [0, 360), we first need to simplify the equation:

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°.