To solve for x, we can use the identities of a right triangle:
cos^2(θ) + sin^2(θ) = 1
Given that sinθ = 2x - 0.5, we can square both sides:
(sinθ)^2 = (2x - 0.5)^2
And given that cosϕ = x + 0.2, we can square both sides:
(cosϕ)^2 = (x + 0.2)^2
Now we can substitute these squared values into the first identity:
(sinθ)^2 + (cosϕ)^2 = 1
(2x - 0.5)^2 + (x + 0.2)^2 = 1
Expand and simplify the equation:
4x^2 - 2x + 0.25 + x^2 + 0.4x + 0.04 = 1
Combine like terms:
5x^2 + 0.4x - 0.71 = 1
Rearrange the equation:
5x^2 + 0.4x - 1.71 = 0
Now we can solve this quadratic equation. Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
a = 5, b = 0.4, c = -1.71
x = (-0.4 ± √(0.4^2 - 4*5*-1.71)) / (2*5)
x = (-0.4 ± √(0.16 + 34.2)) / 10
x = (-0.4 ± √34.36) / 10
x ≈ (-0.4 ± 5.86) / 10
x ≈ 5.46 / 10 or x ≈ (-6.26) / 10
x ≈ 0.546 or x ≈ -0.626
Since x represents a length, it cannot be negative.
Therefore, the value of x is approximately 0.546.
Answer: 0.546
A right triangle has two acute angles, θ and ϕ, so that cosϕ=x+0.2 and sinθ=2x−0.5. What is the value of x? (1 point)
0.7
10
1
0.07
1 answer