A right triangle has two acute angles. θ and Ø, so that cos Ø=x+0.2 and sin θ=2x-0.5. what is the value of x?
the answer is not 0.18
To find the value of x, we can use the Pythagorean identity sin^2θ + cos^2Ø = 1, since the triangle is a right triangle.
Given that cos Ø = x + 0.2 and sin θ = 2x - 0.5, we can substitute these values into the Pythagorean identity:
(2x - 0.5)^2 + (x + 0.2)^2 = 1
Expanding and simplifying:
4x^2 - 2x - x + 0.5^2 + x^2 + 2(0.2)(x) + (0.2)^2 = 1
4x^2 - 3x + 0.25 + x^2 + 0.4x + 0.04 = 1
5x^2 + 0.4x + 0.29 = 1
5x^2 + 0.4x - 0.71 = 0
Now we can solve this quadratic equation for x by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
x = (-0.4 ± √(0.4^2 - 4(5)(-0.71))) / (2(5))
x = (-0.4 ± √(0.16 + 14.2)) / 10
x = (-0.4 ± √14.36) / 10
x = (-0.4 ± 3.79) / 10
Since we are given that x is not equal to 0.18, we discard the positive solution:
x = (-0.4 - 3.79) / 10
x = -4.19 / 10
x = -0.419
Therefore, the value of x is approximately -0.419.