Question

A right triangle has two acute angles, θ 𝜃 and ϕ 𝜙 , so that cosϕ=x+0.2 cos ⁡ 𝜙 = 𝑥 + 0.2 and sinθ=2x−0.5 sin ⁡ 𝜃 = 2 𝑥 − 0.5 . What is the value of x?(1 point) Responses 1 1 0.7 0.7 0.07 0.07 10

Answer 1

In a right triangle, the angles \(\theta\) and \(\phi\) are related by the identity \(\theta + \phi = 90^\circ\). Therefore, we have the relationship:

\[
\cos \phi = \sin \theta
\]

Given:

\[
\cos \phi = x + 0.2
\]
\[
\sin \theta = 2x - 0.5
\]

Setting these equal based on the trigonometric identity:

\[
x + 0.2 = 2x - 0.5
\]

Now, let's solve for \(x\):

1. Rearranging the equation:
\[
0.2 + 0.5 = 2x - x
\]
\[
0.7 = x
\]

So, the value of \(x\) is:

\[
\boxed{0.7}
\]