Question

An angle, theta, is in standard position and its terminal side passes through the point (2,-1). Find the EXACT value of sin theta.

Answer 1

To find the exact value of sin theta when the terminal side passes through the point (2,-1), we need to determine the reference angle and use the property of the sine function in the fourth quadrant.

Step-by-step explanation:

To find the exact value of sin theta, we need to determine the value of theta. In this case, theta is the angle in standard position with its terminal side passing through the point (2,-1). To find theta, we can use the inverse tangent function with the coordinates of the point. Theta = tan^(-1)(-1/2) = -26.57 degrees. Since the angle is in the fourth quadrant, we can use the reference angle of 26.57 degrees to determine the value of sin theta. The sine function is positive in the fourth quadrant, so sin theta = sin 26.57 degrees = 0.445.

Answer 2

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