Question

A right triangle has two acute angles, θ
and ϕ
, so that sinθ=0.3x+0.2
and cosϕ=0.4x+0.4
. Use this information to find the value of x.(1 point)
Responses

0.2
0.2

2
2

−2
negative 2

6

Answer 1

In a right triangle, the angles θ and ϕ satisfy the relationship:

\[
\theta + \phi = 90^\circ
\]

This implies that:

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

Given the equations:

\[
\sin\theta = 0.3x + 0.2
\]
\[
\cos\phi = 0.4x + 0.4
\]

From the relationship between sine and cosine (for complementary angles), we have:

\[
0.3x + 0.2 = 0.4x + 0.4
\]

To solve for \( x \), we rearrange the equation:

\[
0.3x + 0.2 - 0.4x - 0.4 = 0
\]

Combining like terms gives:

\[
-0.1x - 0.2 = 0
\]

Now, we isolate \( x \):

\[
-0.1x = 0.2
\]
\[
x = \frac{0.2}{-0.1} = -2
\]

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

\[
\boxed{-2}
\]