Find the solutions for sec^2x+secx=2 that fit into the interval [0, 2pi]
I figured out that pi/3 (after factoring and switching things around) was a solution, but then I got stuck on cos x = 1/3.
no!
why don't you factor it with the secant as is
sec^2 x + secx - 2 = 0
(secx + 2)(secx-1) = 0
secx = -2 or secx = 1
cosx = -1/2 or cosx = 1
from cosx = -1/2, x = 120º (2pi/3) or 240º (4pi/3)
or
from cosx = 1, x = 0 or 360º (0 or 2pi)
The answers in the back of my book are pi/3,pi, and 5pi/3
To find solutions for the equation sec^2(x) + sec(x) = 2 in the interval [0, 2pi], you have made progress by identifying that x = pi/3 is a solution. Now, let's proceed to find the other solutions.
To continue solving the equation, you started by rearranging it to obtain cos(x) = 1/3. This is correct! Now, to find the values of x that satisfy this equation, we can use the inverse cosine function (also called arccosine).
The arccosine function gives us the angle whose cosine equals a given value. In this case, we want to find the angle x whose cosine equals 1/3. In mathematical notation, we can write:
x = arccos(1/3)
The value of arccos(1/3) can be found using a calculator that has trigonometric functions or by referencing trigonometric identity tables. On most calculators, the arccosine function is usually denoted as "acos" or "cos^(-1)".
To find the value of arccos(1/3) in degrees, you would need to enter "acos(1/3)" into the calculator and convert the result from radians to degrees. The answer is approximately 70.5 degrees.
Now we have found one solution: x = 70.5 degrees (or pi/3 in radians). However, we need to find the other solutions within the interval [0, 2pi].
Since the cosine function is periodic with a period of 2pi, we can find other solutions by adding or subtracting multiples of 2pi to the initial solution. In this case, pi/3 is within the interval [0, 2pi], so we don't need to worry about adding or subtracting 2pi.
Therefore, the solutions for x within the interval [0, 2pi] are x = pi/3 and x = 70.5 degrees (or pi/3 in radians).
Remember to always double-check your solutions by substituting them back into the original equation to ensure they satisfy the equation.