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?
Given the range of -0.25 ≤ x ≤ 0.8, the only option that falls within this range is 0.7. Therefore, the value of x is 0.7.
Since a right triangle has two acute angles, we know that the sum of the measures of the two acute angles must be 90 degrees. Therefore, we have:
θ + ϕ = 90
We are given that cosϕ = x + 0.2. Since the range of the cosine function is [-1, 1], we have:
-1 ≤ x + 0.2 ≤ 1
-1 - 0.2 ≤ x ≤ 1 - 0.2
-1.2 ≤ x ≤ 0.8
Similarly, we are given that sinθ = 2x - 0.5. Since the range of the sine function is [-1, 1], we have:
-1 ≤ 2x - 0.5 ≤ 1
-1 + 0.5 ≤ 2x ≤ 1 + 0.5
-0.5 ≤ 2x ≤ 1.5
-0.5/2 ≤ x ≤ 1.5/2
-0.25 ≤ x ≤ 0.75
From the inequalities -1.2 ≤ x ≤ 0.8 and -0.25 ≤ x ≤ 0.75, we can conclude that:
-0.25 ≤ x ≤ 0.8
Therefore, the value of x is within the range -0.25 to 0.8. We cannot determine the exact value of x without additional information.
options are:
0.07
1
10
0.7