sketch the angle in Theta standard position such that the terminal side of theta has equation for x + 3 y equals 0x less than zero find the valley of sin

sin(θ) = 1/√10 = 0.316

Step-by-step 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

b: |12 – x| ≤ 0.25

step-by-step explanation:      

let x be the possible lengths in inches, of the ribbons jaime cut.

we have been given that jaime is cutting strips of ribbon for a project. he tries to cut each piece to be 12 inches long, but sometimes he cuts it too long or too short by as much as 0.25 inch.  

we can represent this information in an inequality as:

by the definition of absolute value function we will get,

upon combining our inequalities we will get,

therefore, the inequality   describes all the possible lengths (x), in inches, of the ribbons jaime cut.

step-by-step explanation: