Answer:
sin(θ) = 1/√10 = 0.316
Explanation:
Suppose that we have a point (x, y).
The angle between the positive x-axis and a ray that connects the origin with this point (x, y) is called θ, and will be true that:
sin(θ) = y/(√(x^2 + y^2))
cos(θ) = x/(√(x^2 + y^2))
tan(θ) = y/x
Here we want to find the angle θ defined by:
x + 3*y = 0
such that:
x < 0.
If we isolate y, so we get a linear equation:
3*y = -x
y = -x/3 = (-1/3)*x
In the image below you can see the graph of this equation, where we only look at the part where x < 0.
In the sketch, you can see that for any ordered pair with x negative we will get the same angle θ
So we can just evaluate our line with a negative value of x, so we get an ordered pair and we can use one of the above relations to find the value of sin(θ).
If we use x = -3, then:
y = (-1/3)*-3 = 1
Then we have the ordered pair (-3, 1)
So, if we use the sin relation above, we get:
sin(θ) = y/(√(x^2 + y^2)) = 1/( √((-3)^2 + 1^2))
sin(θ) = 1/(√(9 + 1)) = 1/(√10) = 1/√10
sin(θ) = 1/√10 = 0.316