To solve for x, we need to use the properties of a right triangle and the trigonometric identities.
First, let's use the fact that the sum of the squares of the two acute angles in a right triangle is equal to 1:
(cos ϕ)^2 + (sin θ)^2 = 1
Substituting the given values of cos ϕ and sin θ:
(x + 0.2)^2 + (2x - 0.5)^2 = 1
Expanding and simplifying the equation:
x^2 + 0.4x + 0.04 + 4x^2 - 2x + 0.25 = 1
Combining like terms:
5x^2 + 2.25x - 0.71 = 0
Using the quadratic formula, we can solve for x:
x = (-2.25 ± √(2.25^2 - 4(5)(-0.71))) / (2(5))
Calculating the discriminant:
Δ = 2.25^2 - 4(5)(-0.71) = 5.64
Since we are looking for a real value for x, we take the positive square root of the discriminant in the quadratic formula:
x = (-2.25 + √5.64) / (10)
x = (-2.25 + 2.376) / (10)
x = 0.126 / 10
x = 0.0126
Therefore, the value of x is 0.0126, which is approximately equal to 0.01.
A right triangle has two acute angles, θ and ϕ, so that cos ϕ = x + 0.2 and sin θ = 2x−0.5. What is the value of x?(1 point)
10
0.07
1
0.7
1 answer