Let S be the sum of all the possible values of sin x that satisfy the following equation:
5-2cos^2(x)-7sin(x) = 0
S can be written as a/b, where a and b are coprime positive integers. What is the value of a + b?
2cos^2(x)=2-2sin^2(x)
substituting,
5-2+2sin^2(x)-7sin(x) = 0
it comes as
2p^2-7p+2=0 where p=sinx
solving
p=13/4,1/4
as sin(x)<=1
so sin(x)=1/4
a=1
b=4
a+b=5
Eh? Starting at
2p^2 - 7p + 3 = 0
(2p-1)(p-3) = 0
p = 1/2 or 3
but sinx cannot be 3, so
sinx = 1/2 is the only solution
a+b = 1+2 = 3
To find the sum of all the possible values of sin x that satisfy the given equation, we need to solve the equation first.
The equation given is: 5 - 2cos^2(x) - 7sin(x) = 0
Let's rewrite the equation in terms of sin x using the identity: cos^2(x) + sin^2(x) = 1
Replacing cos^2(x) with 1 - sin^2(x), we can rewrite the equation as:
5 - 2(1 - sin^2(x)) - 7sin(x) = 0
Simplifying the equation:
5 - 2 + 2sin^2(x) - 7sin(x) = 0
3 - 7sin(x) + 2sin^2(x) = 0
Now, we have a quadratic equation in terms of sin(x). To solve it, we can factorize it or use the quadratic formula. Since we want to find the sum of all possible values of sin(x), we'll use the quadratic formula.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the equation is:
2sin^2(x) - 7sin(x) + 3 = 0
Comparing it with the quadratic formula, we have:
a = 2, b = -7, c = 3
Substituting the values in the quadratic formula, we get:
sin(x) = [7 ± √(7^2 - 4*2*3)] / (2*2)
sin(x) = [7 ± √(49 - 24)] / 4
sin(x) = [7 ± √25] / 4
Simplifying further, we have two possible values for sin(x):
sin(x) = (7 + 5)/4 = 3/2 or sin(x) = (7 - 5)/4 = 1/2
However, the sine of an angle is always between -1 and 1. So, the solution sin(x) = 3/2 is not valid.
Therefore, the only possible solution for sin x is sin(x) = 1/2.
Now, we can calculate the value of S, the sum of all possible values of sin x:
S = sin(x) = 1/2
So, the value of a + b = 1 + 2 = 3.