The point (2,1) is on the terminal side of an angle in standard position. What are the values of sin theta and cos theta

Answer 1

Hi Gary,

First, we need to plot the point on the coordinate plane and from there we will use that position to create a right triangle that we can use to determine the values of cosine and sine.

From the point (2,1)

x = 2

y = 1

The height of the triangle is the distance from the x-axis to the y coordinate = 1

The length of the triangle is the distance between the origin and the x coordinate = 2

From those 2 values, we can use the Pythagorean theorem to figure out the hypotenuse,

If 1 = a, and 2 = b, and c = the length of the hypotenuse, then,

a² + b² = c²

1² + 2² = c²

5 = c²

√5 = c

The hypotenuse has a length of √5

Now we have all that we need to use the sine and cosine functions

It is important to remember that the angle that we are using for the sine and cosine functions is the angle in standard position which means that it is centered at the origin.

sin(θ) is defined as the opposite over the hypotenuse.

Opposite = height = 1

Hypotenuse = √5

Sin(θ) = 1/√5

If you wanted to get the root out of the radical, you can multiply by √5/√5

1/√5 x √5/√5 = √5/5

Cos(θ) is defined as adjacent over hypotenuse

Adjacent = length = 2

Hypotenuse = √5

Cos(θ) = 2/√5

Once again you can multiply by √5/√5 to get,

2/√5 x √5/√5 = 2√5/5

Let me know if that helped!

Answer 2

cos(θ) = adj / hyp

sin(θ) = opp / hyp

An angle in standard position has its vertex at the origin (0,0) of the x-y coordinate system, the initial side is on the +x-axis, and the terminal side of the angle is drawn by connect the point (0,0) to point (2,1).

If you were to start at the point (2,1), then drop a line straight down to the x-axis, you would see a right-triangle, which will make the problem clearer.

Now the adjacent side of the triangle is length 2, and the opposite side of the triangle is length 1.

adj = 2

opp = 1

To answer the question, determine the length of the hypotenuse, a² + b² = c², solve for c

c = hyp = √5

ANSWER:

cos(θ) = 2/√5 rationalized = (2√5)/5

sin(θ) = 1/√5 rationalized = √5/5