Solve each equation, giving a general formula for all of the solutions:
a. 2sin^2(x)-5 sin(x)-3=0
b. (cos(x)-((sqrt2)/2))(sec(x)-1)=0
just factor the first one, then there will be two values between 0 and 2π, and all multiples of 2π added thereto.
2sin^2(x)-5 sin(x)-3=0
(2sinx+1)(sinx-3) = 0
sinx = -1/2
sinx = 3 (not a valid solution)
The other is already factored:
(cos(x)-((√2)/2))(sec(x)-1)=0
(cosx-1/√2)(secx-1) = 0
cosx = 1/√2
secx = 1
a. To solve the equation 2sin^2(x)-5sin(x)-3=0, we can use factoring technique or the quadratic formula.
First, let's make a substitution by letting sin(x) = t. Now the equation becomes 2t^2 - 5t - 3 = 0.
To factor the quadratic equation, we need to find two numbers whose product is -6 (2*-3) and whose sum is -5. We can write this quadratic equation as (2t + 1)(t - 3) = 0 by factoring.
Now we have two equations: 2t + 1 = 0 or t - 3 = 0.
Solving the first equation, 2t + 1 = 0, we get t = -1/2.
Solving the second equation, t - 3 = 0, we get t = 3.
Since we substituted sin(x) = t, we can substitute those values back:
For sin(x) = -1/2, the corresponding angles fall in the second and fourth quadrants. The solutions are x = arcsin(-1/2) + 2πn and x = π - arcsin(-1/2) + 2πn, where n is an integer.
For sin(x) = 3, there are no solutions because the value of sin(x) cannot exceed 1.
Therefore, the general formulas for all solutions of the equation 2sin^2(x)-5sin(x)-3=0 are x = arcsin(-1/2) + 2πn and x = π - arcsin(-1/2) + 2πn, where n is an integer.
b. To solve the equation (cos(x)-((√2)/2))(sec(x)-1) = 0, we can set each factor equal to zero and solve for x separately.
Setting cos(x) - (√2)/2 = 0, we find the corresponding angles fall in the first and fourth quadrants. The solutions are x = arccos(√2/2) + 2πn and x = -arccos(√2/2) + 2πn, where n is an integer.
Now, setting sec(x) - 1 = 0, we can rewrite it as sec(x) = 1. Since sec(x) is the reciprocal of cos(x), we can also rewrite it as cos(x) = 1/1, which is just cos(x) = 1.
The angle where cos(x) = 1 is x = 0.
Therefore, the general formulas for all solutions of the equation (cos(x)-((√2)/2))(sec(x)-1) = 0 are x = arccos(√2/2) + 2πn, x = -arccos(√2/2) + 2πn, and x = 0, where n is an integer.