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

6
6

2
2

−2

Answer 1

To solve for \(x\) using the given equations related to a right triangle, we start with the definitions:

1. Since \( \sin \theta + \cos \phi = 1 \), we can use the identities of sine and cosine to find the relationship between the angles.

Given:
- \( \sin \theta = 0.3x + 0.2 \)
- \( \cos \phi = 0.4x + 0.4 \)

Since \( \theta \) and \( \phi \) are complementary angles, we know that:

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

Setting the two equations equal to each other:

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

Now, we'll solve for \(x\):

1. Rearrange the equation:
\[
0.3x - 0.4x = 0.4 - 0.2
\]

2. Combine like terms:
\[
-0.1x = 0.2
\]

3. Divide by \(-0.1\):
\[
x = \frac{0.2}{-0.1} = -2
\]

So, the value of \(x\) is \(-2\).

Thus, the final answer is:
\[
\boxed{-2}
\]