Given that 3cos2x + 17sinx = 8, find the exact value of sinx .
3cos2x + 17sinx - 8 = 0
3(1-2sin^2x) + 17sinx - 8 = 0
6sin^2x - 17sinx + 5 = 0
(3sinx-1)(2sinx-5) = 0
sinx = 1/3 or 2/5
cos(2x) = 1 – 2 sin^2(x)
substituting ... 3 - 6 sin^2(x) + 17 sin(x) = 8
... -6 sin^2(x) + 17 sin(x) - 5 = 0
... 6 sin^2(x) - 17 sin(x) + 5 = 0
factoring ... [3 sin(x) - 1] [2 sin(x) - 5] = 0
3 sin(x) - 1 = 0 ... sin(x) = 1/3
2 sin(x) - 5 = 0 ... sin(x) = 5/2 ... unrealistic solution
Why did the sine go to the therapist? Because it had too many unresolved cosine issues!
To find the exact value of sinx, we can rewrite the equation 3cos2x + 17sinx = 8 using the double angle formula for cosine.
The double angle formula for cosine is cos2x = 2cos^2(x) - 1.
Substituting this into our equation, we have:
3(2cos^2(x) - 1) + 17sinx = 8
Simplifying further:
6cos^2(x) - 3 + 17sinx = 8
6cos^2(x) + 17sinx = 11
Since we want to find the value of sinx, we can use the Pythagorean identity for sine and cosine:
sin^2(x) + cos^2(x) = 1
Rearranging this equation, we have:
sin^2(x) = 1 - cos^2(x)
Substituting this into our previous equation, we get:
6(1 - sin^2(x)) + 17sinx = 11
6 - 6sin^2(x) + 17sinx = 11
Rearranging further, we have a quadratic equation:
6sin^2(x) - 17sinx + 5 = 0
Now, let's solve this quadratic equation using factorization or the quadratic formula.
To find the exact value of sinx, we can first manipulate the given equation to get sinx in terms of cosx.
Let's start by using the double angle identity for cosine:
cos2x = 2cos^2x - 1
Substituting this into the given equation, we have:
3(2cos^2x - 1) + 17sinx = 8
Expanding and rearranging the equation, we get:
6cos^2x - 3 + 17sinx = 8
6cos^2x + 17sinx - 11 = 0
Now, we can use the trigonometric identity sin^2x + cos^2x = 1 to express cos^2x in terms of sinx:
cos^2x = 1 - sin^2x
Substituting this into the equation above, we get:
6(1 - sin^2x) + 17sinx - 11 = 0
6 - 6sin^2x + 17sinx - 11 = 0
-6sin^2x + 17sinx - 5 = 0
Now, we have a quadratic equation in terms of sinx. We can solve this equation using factoring, completing the square, or the quadratic formula. In this case, factoring is not straightforward, so we will use the quadratic formula.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Applying the quadratic formula to our equation, we have:
sinx = (-17 ± √(17^2 - 4(-6)(-5))) / (2(-6))
sinx = (-17 ± √(289 - 120)) / (-12)
sinx = (-17 ± √169) / (-12)
sinx = (-17 ± 13) / (-12)
We have two possible solutions:
1. sinx = (-17 + 13) / (-12) = -4/12 = -1/3
2. sinx = (-17 - 13) / (-12) = -30/(-12) = 5/2
However, the values of sine are limited to the range -1 to 1. Therefore, the only valid solution is:
sinx = -1/3