Answer:
sinѲ = -1/√5 or -√5/5
Step-by-step explanation:
On the Cartesian plane, we have radius (r) and angle (Ѳ) theta.
On the Cartesian plane we also have x, y where:
x = r cosѲ .......Equation 1
and y = r sinѲ ........Equation 2
r = √x² + y²
In the question we are given points (2,-1)
Where x = 2 , y = -1
We would solve for r by substituting 2 for x and -1 for y
r = √ 2² + -1²
r = √ 4 + 1
r = √5
In the question, we were asked to find what sin theta (Ѳ) is. Hence, we would be substituting √5 for r in Equation 2
y = r sinѲ
Where y = -1 and r = √5
-1 = √5 sinѲ
Divide both sides by √5
sinѲ = -1/√5
We can also represent sin Ѳ in a proper form, by multiplying both top and bottom by √5
sinѲ = -√5/5
Therefore, sinѲ = -1/√5 of -√5/5
The ray of an angle not on the x-axis is called the terminal side.
The exact value of [tex]\sin(\theta)[/tex] is [tex]-\frac{\sqrt 5}{5}[/tex]
The point is given as: (x,y) = (2,-1)
Where
[tex]r^2 = x^2 + y^2[/tex]
Take square roots of both sides:
[tex]r =\sqrt{ x^2 + y^2[/tex]
Substitute values for x and y
[tex]r =\sqrt{ 2^2 + (-1)^2[/tex]
[tex]r =\sqrt{ 4 + 1[/tex]
[tex]r =\sqrt 5[/tex]
Also, we have the following equation on a Cartesian plane
[tex]y = r \sin(\theta)[/tex]
Substitute values for y and r
[tex]-1 = \sqrt 5 \times \sin(\theta)[/tex]
Divide both sides by [tex]\sqrt 5[/tex]
[tex]-\frac{1}{\sqrt 5} = \sin(\theta)[/tex]
Rewrite as:
[tex]\sin(\theta) = -\frac{1}{\sqrt 5}[/tex]
Rationalize
[tex]\sin(\theta) = -\frac{1}{\sqrt 5} \times \frac{\sqrt 5}{\sqrt 5}[/tex]
[tex]\sin(\theta) = -\frac{\sqrt 5}{5}[/tex]
Hence, the exact value of [tex]\sin(\theta)[/tex] is [tex]-\frac{\sqrt 5}{5}[/tex]
Read more about terminal positions at:
https://brainly.com/question/15384896
