To solve the quadratic equation 10cos^2x - 17sinx - 16 = 0, we need to use trigonometric identities to rewrite it in terms of one trigonometric function.
First, recall the Pythagorean identity: sin^2x + cos^2x = 1. Rearranging this equation gives: cos^2x = 1 - sin^2x.
Now substitute cos^2x with 1 - sin^2x in the given equation: 10(1 - sin^2x) - 17sinx - 16 = 0. Expand and simplify: 10 - 10sin^2x - 17sinx - 16 = 0. Rearrange the terms: -10sin^2x - 17sinx - 6 = 0.
Now we have a quadratic equation in terms of sinx. Let y = sinx, the equation becomes: -10y^2 - 17y - 6 = 0.
Solve this quadratic equation for y using the quadratic formula: y = (-(-17) ± √((-17)^2 - 4(-10)(-6))) / (2*(-10)).
y = (17 ± √(289 + 240)) / -20. y = (17 ± √529) / -20. y = (17 ± 23) / -20.
Now we have two possible solutions for y: y = (17 + 23) / -20 = 40 / -20 = -2, or y = (17 - 23) / -20 = -6 / -20 = 0.3.
Since we assumed y = sinx, we need to find the corresponding values of x for each solution. For y = -2, there is no real solution for sinx since -1 ≤ sinx ≤ 1. For y = 0.3, we can find the corresponding angle by sinx = 0.3. This implies x = arcsin(0.3) ≈ 0.3047 radians.
Therefore, the solution to the equation 10cos^2x - 17sinx - 16 = 0 is x ≈ 0.3047 radians.
To solve the quadratic equation 10cos^2x - 17sinx - 16 = 0, we need to use trigonometric identities to rewrite it in terms of one trigonometric function.
First, recall the Pythagorean identity:
sin^2x + cos^2x = 1.
Rearranging this equation gives:
cos^2x = 1 - sin^2x.
Now substitute cos^2x with 1 - sin^2x in the given equation:
10(1 - sin^2x) - 17sinx - 16 = 0.
Expand and simplify:
10 - 10sin^2x - 17sinx - 16 = 0.
Rearrange the terms:
-10sin^2x - 17sinx - 6 = 0.
Now we have a quadratic equation in terms of sinx. Let y = sinx, the equation becomes:
-10y^2 - 17y - 6 = 0.
Solve this quadratic equation for y using the quadratic formula:
y = (-(-17) ± √((-17)^2 - 4(-10)(-6))) / (2*(-10)).
y = (17 ± √(289 + 240)) / -20.
y = (17 ± √529) / -20.
y = (17 ± 23) / -20.
Now we have two possible solutions for y:
y = (17 + 23) / -20 = 40 / -20 = -2, or
y = (17 - 23) / -20 = -6 / -20 = 0.3.
Since we assumed y = sinx, we need to find the corresponding values of x for each solution.
For y = -2, there is no real solution for sinx since -1 ≤ sinx ≤ 1.
For y = 0.3, we can find the corresponding angle by sinx = 0.3.
This implies x = arcsin(0.3) ≈ 0.3047 radians.
Therefore, the solution to the equation 10cos^2x - 17sinx - 16 = 0 is x ≈ 0.3047 radians.