Calculating Arc Length, Triangle, and Sector Areas with Theta

this video is about arc length triangle area and a sector areas for

number one the diagram shows a circle center o with radiance four centimeters

point a and a b lie on the circumference of the circle and an angle aob equals a theta

where this theta is between 0 to 180 degree find the area of the shaded area in

terms of theta

look at the one a we are looking for the shaded area

for this shaded area if we can figure out the sector area then you subtract the triangle area you

will get this shaded area therefore the

sector area equals theta over

360 degree times pi r squared

r is a 4 squared so we get

theta over 360 degree times 16 pi

reduce you will get this is a 2 this is a 45

so we get that 2 pi theta over

45 degree for the triangle area equals one half

a b that is a four times a four because

this is a 4 then sine theta

so we get 8 sine theta therefore the shaded area a equals

2 pi theta over 45 degree

minus 8 sine theta

for 1b the area of what shaded the region is a 12

centimeter square find the value of theta so we will set up

based 2 pi theta over 45 degree minus 8

sine theta equals 12. i will show you use graphing calculator

to get the answer if you only have one solution you will use a

method solver before you go to method third world you have to change into equal to zero

form so i will move this 12 to the left side

i will get 2 pi theta over 45 degree minus 8

sine theta minus 12 equal to zero now i will show you use graphing calculator how to get

this theta in degree first of all you need to clear the

memory second plus seven one

two then go to mode we need a degree then we go to y equals we will get alpha web equals get a

fraction then 2 pi

x over 45 remember for every variable you need

to change into x when you use graphing calculator to get the solution then minus

8 sine x then minus the 12

then we go to quit math use this upper arrow to get a

solver enter then we will put y1 here y1 is the alpha

trees you will get y1 then enter

for this x equals you can put in any number you like

your favorite number for example my theory number is a 7. then i do alpha

enter that's a solve the answer will be 1 30 degree

so answer will be 1 30 degree now let's go to

2a the following diagram shows a right angled triangle abc with ac equals ten

a b equals six b c equals eight the points d and f

lie on ac bd is a perpendicular to ac before is the arc of a circle

centered at a in another words we have af4 equals six because this is the circle

bd is perpendicular to ac the region r is bounded by b d

d f and the arc b e f find the angle

b a c since this is a right triangle we know

all these are three sides therefore we can use the sukhan torah to get this angle

b ac let's label as a theta corresponding to theta we know this is a opposite side

this is a adjacent side here's a hypotenuse side

i can use the sine theta equals eight over ten then theta equals

sine inverse sine of eight over ten

then we go to graphing calculator answer is a 53.1 degree

for 2b find the area of r for this area we can get

by doing subtraction so

for the sector area that equals theta over

360 degree times the pi r squared r is a six square

so we get theta is a 53.5 over

360 degree times 36

pi then i need to figure out the triangle of

a d b this one for this right triangle

i have two ways to figure this out i can use one half for base times height also i can use

the area formula one half a b sine c anyway

you will choose which one you got to use i will choose one half of a b sin c

so i need to figure out this a d

for this a d corresponding to theta will be adjacent and a b will be

hypotenuse using so

can torah we're going to use a cosine so we have a cosine

theta equals a over h that's a d

over a b is a six so cross multiply you will get the a d equals six times the cosine

theta so we can get

the area of this triangle that will be

area of triangle a d b

equals one half a will be six

b will be a d then sine theta

so we plug in you will get one half

six eighty will be six times the cosine theta

for sine theta i will bring down according to this diagram

sine theta equals 8 over 10 and a cosine theta that equals

6 over 10. or you can just plug in this degree this is really up to you so we get this

equals one half times 36 times the 6 over 10

times 8 over 10. the area of for shaded region

equals 53.1 over 360

then times the 36 pi minus one half times 36 times 6 over 10 times

8 over 10. answer will be 8.01

centimeter squared let's go to 3a oab is a sector of the circle with a central o and a

radiance r as shown in the following diagram the angle aob is a theta

where theta is between 0 to 90 degree the point c lies on o a and o a is perpendicular to b c

show that oc equals r cosine theta in right triangle o b c

we can use uh so kantova label this is a o

here's a a this one is h for cosine theta

equals a over h that is a

oc over ob

that is r then we do cross multiply you will get the o say equals

r times the cosine theta whenever you say show that which means

you just need to work this out show the work

for the 3b find the area of triangle o b

c in terms of r and the

theta we know the area of this triangle we use one half of a b

sine c one half a b sin c for

this theta it's an included angle you are supposed to use the ob and also oc so i can write down one half is the ob

times the oc then times the sine theta for ob we know that is a r

o say is r times a cosine theta sine theta

i organized that is one half r squared cosine theta sine theta that's for 3b

for 3c given that the area of triangle o b c

is a three over five of the area of our sector

o a b find theta this is a three c

so for the sector area that equals theta over 360 degree

times pi r squared so pi r that's ob so i just write down

pi r squared then the triangle area is this one

then according to this area of a triangle o b c

is a 3 over 5 of area of sector o a b i can write down

that one half r squared cosine theta sine theta equals

3 over 5 times this sector area

that's a theta over 360 degree times the pi r squared before

you look for this theta we can reduce this is r square r squared is gone then we will use a

math solver to work this out move everything to the left side i have one half

cosine theta sine theta minus three over five times the theta over

360 degree pi that equal to zero

i will show you how to use graphing calculator get this after you clear the memory remember the

mode is a degree then go to y equals

you will put in 1 divided by 2 sine x parenthesis then cosine x

parenthesis then subtract 3 divided by 5

times the x divided by 360 degree

times the pi then we go to quit

math upper arrow go to solver you still need to put a way one so alpha

trees y1 then enter

you can put your favorite number i put a 7

then i do alpha enter they give me zero i know zero is not my answer because the

theta cannot be zero and when you look at the problem this

theta is between 0 to 90 so let me put 90 close to 90 so i do alpha

enter you will get the 47.6 degree

if you are not familiar with the solver you can go to graph

and uh figure out the x-intercept

so i will push a graph nothing happens here

because the domain is a too small i will change the domain for x

this set is between 0 to 90 so i change it into 0 to 90. and

this is a skill normally i use the maximum number divided by 10. let me do 9

then i graph again you will see something like that where you see this a graph or that means

your y axis is too big so let me go to window

the y axis i will do negative one and uh here is a maximum let's do one so we graph again

this is a little bit clear then i do the second trace 2

that's zero so i move to the left of this is the intersection i do enter then

do the right bound i will do over here so enter then enter you will get the same

answer so it's really up to you how to figure out the solution by graphing calculator either by graph or

by mass solver let's go to number four find the value

of the included acute angle of a triangle width size

of left 17 centimeters and a 23 centimeters and an area is a 75

centimeter squared we know this formula area equals one half of a b

sine c then we plug in the value area is 75

one half a will be 17 b will be 23 then is the sun c we're looking for this angle c now

first of all we need to isolate the sign saying on the right side

therefore we will multiply by 2

both sides so that reduce here then divided by

17 both sides then divided by 23 both sides

so that we will get 2 times 75 over 17 times 23 equals sine c

enough c equals inverse sine of

this 2 times 75 over 17 times

23 then equals let's go to graphing calculator

first of all you need to check the mode it's a degree then you will do second sign

alpha y equals enter 2 times 75

over 23 times 17

then parenthesis back enter you will get the answer 22.6 therefore the angle say

is a 22.6 degree let me show you use a ukraine solver to

do this so we have a 75

equals one half 17 times a 23

then sine c while using ukraine solver to solve this

equation make sure you have to get equal to zero therefore we will move 75 or

to the right side you will get the zero on the left side one half

17 times a 23 sine c minus 75.

we know this is a acute angle this is an acute angle acute angle means this say is between

0 to 90 degree so

in the ukraine solver when you estimate the value try to use the value between 0 to 90

degree make sure your mode is a degree

then you go to y equals we will plug in here 1 divided by 2 times

17 times 23 then sine

x all variable you need to tune into x then minus 75

then we go to math upper arrow go to solver

alpha trace we get way one then enter

since we need an acute angle using 45 degree or you can use a 30

just any acute angle over here then you do the alpha

enter to solve your answer will be 22.6 you see you will get the same answer [Music]