Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

A point starts at the location
(2,0) and moves counterclockwise along a circular path with a radius 2 units long centered at the origin of an
xy plane. An angle with its vertex at the circle's center measures
theta radians and subtends the path the point travels. Let
x represent the point's
x coordinate. (Draw a diagram of this to make sure you understand the context!)
a. Complete the following statements.

As
theta varies from 0 to
(pi)/(2),x varies from to units.
As
theta varies from
(pi)/(2) to
pi,x varies from to units.
As
theta varies from
pi to
(3pi)/(2),x varies from to units.
As
theta varies from
(3pi)/(2) to
2pi,x varies from to units.

A point starts at the location $(2,0)$ and moves counterclockwise along a circular path with a radius $2$ units long centered at the origin of an $x y$ plane. An angle with its vertex at the circle's center measures $\theta$ radians and subtends the path the point travels. Let $x$ represent the point's $x$ coordinate. (Draw a diagram of this to make sure you understand the context!) $\newline$ a. Complete the following statements. $\newline$ - As $\theta$ varies from $0$ to $\frac{\pi}{2}, x$ varies from to units. $\newline$ - As $\theta$ varies from $\frac{\pi}{2}$ to $\pi, x$ varies from to units. $\newline$ - As $\theta$ varies from $x y$ $1$ to $x y$ $2$ varies from to units. $\newline$ - As $\theta$ varies from $x y$ $4$ to $x y$ $5$ varies from to units.

View step-by-step help

View step-by-step help

Write equations of cosine functions using properties

Full solution

Q. A point starts at the location

(2,0)

and moves counterclockwise along a circular path with a radius

2

units long centered at the origin of an

x y

plane. An angle with its vertex at the circle's center measures

\theta

radians and subtends the path the point travels. Let

x

represent the point's

x

coordinate. (Draw a diagram of this to make sure you understand the context!)

\newline

a. Complete the following statements.

\newline

- As

\theta

varies from

0

to

\frac{\pi}{2}, x

varies from to units.

\newline

- As

\theta

varies from

\frac{\pi}{2}

to

\pi, x

varies from to units.

\newline

- As

\theta

varies from

x y

1

to

x y

2

varies from to units.

\newline

- As

\theta

varies from

x y

4

to

x y

5

varies from to units.

Visualize Problem: First, let's visualize the problem by drawing a circle with radius $2$ units centered at the origin of an $xy$ -plane. A point starts at $(2,0)$ and moves counterclockwise along the circle. The $x$ -coordinate of the point is given by the cosine of the angle $\theta$ , since the point is on a circle of radius $2$ , the $x$ -coordinate is $2 \cdot \cos(\theta)$ .
Theta from $0$ to $\frac{\pi}{2}$ : As $\theta$ varies from $0$ to $\frac{\pi}{2}$ , the $x$ -coordinate of the point varies from $2$ to $0$ units. This is because $\cos(0) = 1$ and $\cos\left(\frac{\pi}{2}\right) = 0$ , and since the radius is $2$ , we multiply these values by $2$ .
Theta from $\frac{\pi}{2}$ to $\pi$ : As $\theta$ varies from $\frac{\pi}{2}$ to $\pi$ , the $x$ -coordinate of the point varies from $0$ to $-2$ units. This is because $\cos\left(\frac{\pi}{2}\right) = 0$ and $\cos(\pi) = -1$ , and since the radius is $\pi$ $0$ , we multiply these values by $\pi$ $0$ .
Theta from $\pi$ to $\frac{3\pi}{2}$ : As $\theta$ varies from $\pi$ to $\frac{3\pi}{2}$ , the x-coordinate of the point varies from $-2$ to $0$ units. This is because $\cos(\pi) = -1$ and $\cos\left(\frac{3\pi}{2}\right) = 0$ , and since the radius is $2$ , we multiply these values by $2$ .
Theta from $(3\pi)/(2)$ to $2\pi$ : As $\theta$ varies from $(3\pi)/(2)$ to $2\pi$ , the x-coordinate of the point varies from $0$ to $2$ units. This is because $\cos((3\pi)/(2)) = 0$ and $\cos(2\pi) = 1$ , and since the radius is $2$ , we multiply these values by $2$ .

More problems from Write equations of cosine functions using properties

Question

Find an equation for a sinusoidal function that has period

2\pi

1

, and contains the point

\left(\pi, -1\right)

\newline

Write your answer in the form

f(x)=A\cos(Bx+C)+D

A

B

C

D

are real numbers.

\newline

f(x)=

Posted 1 month ago

Question

Find an equation for a sinusoidal function that has period

2\pi

1

, and contains the point

\left(-\frac{\pi}{2},-1\right)

. Write your answer in the form

f(x) = A\sin(Bx + C) + D

A

B

C

D

are real numbers. `f(x)` =______

Posted 2 months ago

Question

Find an equation for a sinusoidal function that has period

2\pi

1

, and contains the point

(0,-1)

\newline

Write your answer in the form

f(x) = A\cos(Bx + C) + D

A

B

C

D

are real numbers.

\newline

f(x) =

Posted 2 months ago

Question

Franklin Middle School is having a canned food drive. In Mr. Lao's class, Mr. Lao brought

10

cans of food, and each student brought

2

cans. In Ms. Roger's class, Ms. Rogers did not bring any cans, but each student brought

3

cans of food. Write two equations in slope-intercept form to represent the number of cans of food for each class

y

in terms of the number of students

x

Posted 2 months ago

Question

Franklin Middle School is having a canned food drive. In Mr. Lao's class, Mr. Lao brought

10

cans of food, and each student brought

2

cans. In Ms. Roger's class, Ms. Rogers did not bring any cans, but each student brought

3

cans of food. Write two equations in slope-intercept form to represent the number of cans of food for each class

y

in terms of the number of students

x

Posted 2 months ago

Question

Kris is wrapping Christmas lights around the railing that runs around three sides of his square porch If one side of his porch is

12

feet long, and each strand of his Christmes lights will wrap around a

70

-inch length of the railing, what is the minimum number of strands Kris needs to completely wrap the porch railing?

\newline

6

\newline

7

\newline

8

\newline

9

Posted 2 months ago

Question

Kris is wrapping Christmas lights around the railing that runs around three sides of his square porch If one side of his porch is

12

feet long, and each strand of his Christmas lights will wrap around a

70

-inch length of the railing, what is the minimum number of strands Kris needs to completely wrap the porch railing?

\newline

6

\newline

7

\newline

8

\newline

9

Posted 2 months ago

Question

For a given input value

b

f

outputs a value

a

to satisfy the following equation.

\newline

4a+7b=-52

\newline

Write a formula for

f(b)

b

\newline

f(b)=\square

Posted 2 months ago

Question

For a given input value

x

g

outputs a value

y

to satisfy the following equation.

\newline

-4x-6=-5y+2

\newline

Write a formula for

g(x)

x

\newline

g(x)=\square

Posted 4 months ago

Question

For a given input value

a

f

outputs a value

b

to satisfy the following equation.

\newline

-3a+6b=a+4b

\newline

Write a formula for

f(a)

a

\newline

f(a)=\square

Posted 4 months ago