SOLUTION: Solving the following equation will require you to use the quadratic formula. Solve the equation for θ between 0° and 360°, and round your answers to the nearest tenth of a de

Algebra ->  Trigonometry-basics -> SOLUTION: Solving the following equation will require you to use the quadratic formula. Solve the equation for θ between 0° and 360°, and round your answers to the nearest tenth of a de      Log On


   

Question 1008893: Solving the following equation will require you to use the quadratic formula. Solve the equation for θ between 0° and 360°, and round your answers to the nearest tenth of a degree.
2sin^2θ = 3−4cosθ
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
problem is:

2sin^2(theta) = 3 - 4cos(theta).

since sin^2(theta) = 1 - cos^2(theta), substitute for sin^2(theta) to get:

2 * (1 - cos^2(theta) = 3 - 4cos(theta)

simplify to get:

2 - 2cos^2(theta) = 3 - 4cos(theta)

add 2cos^2(theta) and subtract 2 from both sides of the equation to get:

0 = 3 - 4cos(theta) + 2cos^2(theta) - 2

combine like terms and flip the equation to get:

2cos^2(theta) - 4cos(theta) + 1 = 0

that's your quadratic equation that you need to factor.

use the quadratic formula to factor it as follows:

since the equation is in standard form of ax^2 + bx + c = 0, you get:

a = 2
b = -4
c = 1

the quadratic formula is: 

cos(theta) = -b plus or minus sqrt(b^2 - 4ac)
             --------------------------------
                            2a


replace a,b,c with their values and you get: 

cos(theta) = -(-4) plus or minus sqrt((-4)^2 - 4*2*1)
             --------------------------------
                            2*2


you will get:

cos(theta) = (4 + sqrt(8))/4
or:
cos(theta) = (4 - sqrt(8))/4

the result will be:

cos(theta) = 1.707106781
or:
cos(theta) = .2928932188

cosine can't be greater than 1, so the solution is:

cos(theta) = .2928932188

solve for theta to get:

theta = 72.96875154 degrees.

that would be in quadrant 1.

cosine is positive in quadrants 1 and 4 only.

your angle is therefore in quadrant 1 and in quadrant 4.

the angle in quadrant 4 is 360 - 72.96875154 = 287.0312485 degrees.

your solution is that theta = 72.96875154 or 287.0312485.

round that to a tenth of a degree and your solution is:

theta = 73.0 or 287.0 degrees.

the following picture shows the solution graphically.

2 equations were graphed.

they are:

y = sin^2(theta)

y = 3 - 4cos(theta)

the intersection of the 2 equations on the graph is when their values are equal.

that occurs at theta = 73 and theta = 287 between 0 and 360 degrees.

$$$

the graphical solution conforming to the algebraic solution is shown below:

this graph is of the equation y = 2cos^2(theta) - 4cos(theta) + 1

the solution, in this case, is when the graph crosses the x-axis.

$$$