how do you solve: find cos2x if sinx is equal to 1/5
cos2x if sinx = 1/5
(1-2sin^2 x)(sinx) = 1/5
sinx - 2 sin^3 x = 1/5
5sinx - 10sin^3 x = 1
10sin^3 x - 5sinx + 1 = 0
Now you have a nasty cubic to solve
I use this on-line cubic equation solver
http://www.1728.com/cubic.htm
it gave me
sinx = .56959
sinx = -.79143
sinx = .22183
each of those will give you 2 different solutions
I will do the first
if sinx = .56959 , x is in quadrants I or II and
x = 34.72° or x = 145.28°
there will be 6 answers between 0 and 360
To solve for cos2x given that sinx is equal to 1/5, we can use the trigonometric identity:
cos^2x + sin^2x = 1
Let's substitute sinx with its given value:
cos^2x + (1/5)^2 = 1
Simplifying the equation, we have:
cos^2x + 1/25 = 1
Subtracting 1/25 from both sides:
cos^2x = 1 - 1/25
cos^2x = 24/25
To determine the value of cos2x, we can use another trigonometric identity:
cos2x = cos^2x - sin^2x
Substituting the value of cos^2x from the previous calculation:
cos2x = 24/25 - (1/5)^2
cos2x = 24/25 - 1/25
cos2x = 23/25
Therefore, cos2x is equal to 23/25.