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) Responses 10 10 0.07 0.07 1 1 0.7

Answer 1

Using the given information, we can set up two equations:

cosϕ = x + 0.2 ----- Equation (1)
sinθ = 2x - 0.5 ----- Equation (2)

To solve for x, we need to find ϕ and θ.
Since ϕ is an acute angle, cosϕ will be positive. From Equation (1), we have:

x + 0.2 > 0
x > -0.2

Now let's solve for ϕ using Equation (1):
cosϕ = x + 0.2
ϕ = cos^(-1)(x + 0.2)

Similarly, sinθ will be positive since θ is also an acute angle. From Equation (2), we have:

2x - 0.5 > 0
2x > 0.5
x > 0.25

Now let's solve for θ using Equation (2):
sinθ = 2x - 0.5
θ = sin^(-1)(2x - 0.5)

To find the value of x, we need to find the intersection point of the two ranges of x that we found earlier:
x > -0.2
x > 0.25

The smallest number that satisfies both of these inequalities is x = 0.25.

Therefore, the value of x is 0.25.