Question

How to solve 2(sin(x))^2 – 5 sin(x) = 3?

Answer 1

make a u substitution

u= sin(x)

what would your equation be?

Answer 2

sin^-1(sin(x)) = sin^1(x) ?

Answer 3

x = sin^1(y)

Answer 4

sin^-1(sin(x)) =x

no idea what you're doing

u substitution,
that means

sin(x) =u
and
(sin(x))^2 =u^2

so your equation would become
2u^2-5u=3

Answer 5

How do you find sin(x)?

Answer 6

once you make the u substitution, move the 3 over to the right side, then you have a quadratic equation

solve for u

once you have solved for u, you substitute sin(x) back in for u

then you solve for x

Answer 7

So a = 2, b = -5, and c = -3 in the quadratic equation?

Answer 8

yes, but you should probably try factoring it first

always use the quadratic equation as a last resort

Answer 9

I don't know how to factor and got an imaginary number in the quadratic equation

Answer 10

Nevermind I think I made a mistake

Answer 11

basically look at "a", then write down all the multiple pairs for a

a = 2
so the multiple pairs would be [1,2] and [-1,-2]

now look at c, and write down all the multiple pairs

Answer 12

c= -3
so the multiple pairs would be?

Answer 13

factor pairs, multiple pairs, same thing

Answer 14

12 and -.5?

Answer 15

i see random numbers and have no idea what they mean

Answer 16

Is this the factor pair?

Answer 17

12 * -.5 =?

Answer 18

-6, what is this for?

Answer 19

its not a multiple pair of -3

Answer 20

i also doubt those are the roots (solution of u)

Answer 21

a = 2
so the multiple pairs would be [1,2] and [-1,-2]
now look at c,
c= -3

what would the multiple pairs be?

Answer 22

I got 3/2 and 1 for the roots, thats how far I've gotten in understanding

Answer 23

Would the multiple pairs be (1,-3) and (-1,3)?

Answer 24

yes they would be

now are you familiar with dot product?

Answer 25

After figuring out the multiple pairs for a-c what do I do?

Answer 26

basically, you need to find some combination of the the multiples pairs of a and c, whose dot product is equal to b

so using a= [1 2] and c= [ 1 -3]

the dot product would be (1*1) +(2*-3)
1-6 = -5
b=-5

thus meaning the equation would be (2u+1)(u-3) = 2u^2-5u-3=0

Answer 27

anyways

since you've already found the roots

u = -1/2 and u =3

we made the u= sin (x)

so we substitute that back to get

sin(x) = -1/2 & sin(x) =3

solve for x

Answer 28

x = -.52 and...

Answer 29

yea, you'd have to use a calculator for that part

and then there becomes an issue about whether x is in degrees or radians

Answer 30

I was told in radians because the answer needs to be in pi

Answer 31

I can't figure out sin^-1(3)

Answer 32

well, then make sure your calculator is set to radians

and in order to determine the ratio to pi, then just divide by pi, and put that number infront of pi

Answer 33

And why is sin^-1(3) an error?

Answer 34

because sin(x) can only be between -1 and 1

Answer 35

thus sin(x)= 3 doesnt make sense

Answer 36

So only one solution?

Answer 37

yup

Answer 38

Awesome thank you!

Answer 39

no prob