9. a circle has an arc of length 56pi that is intercepted by a central angle of 120 degrees. what is the radius of the circle?

10. the terminal side of (0) passes through the point (11,-9). what is the exact value of sin(0) in simplified form?

Part 4)

Part 9)

Part 10)

Step-by-step explanation:

Part 4) A circle has an arc of length 56pi that is intercepted by a central angle of 120 degrees. What is the radius of the circle?

we know that

The circumference of a circle subtends a central angle of 360 degrees

The circumference is equal to

using proportion

simplify

solve for r

Part 9) Given cos(∅)=-2/3 and ∅ lies in Quadrant III. Find the exact value of sin(∅) in simplified form

Remember the trigonometric identity

we have

substitute the given value

square root both sides

we know that

If ∅ lies in Quadrant III

then

The value of sin(∅) is negative

Part 10) The terminal side of ∅ passes through the point (11,-9). What is the exact value of sin(∅) in simplified form?    

see the attached figure to better understand the problem

In the right triangle ABC of the figure

Find the length side AC applying the Pythagorean Theorem

substitute the given values

so

simplify

Remember that      

The point (11,-9) lies in Quadrant IV

then      

The value of sin(∅) is negative

therefore

ab will be equal to