Question

Write as an algebraic expression in x:
sin( arctan(3/4) )

Answer 1

this isn't something that varies
it is just a number
do you mean rewrite sin(arctan(x)) as an algebraic expression in terms of x?

Answer 2

then use that algebraic expression to evaluate sin(arctan(x)) for x=3/4

Answer 3

anyways to write sin(arctan(x))
as an algebraic expression
let u=arctan(x)
so tan(u)=x
draw a right triangle then
use that triangle to find sin(u) in terms of x

Answer 4

sin(u)=3

Answer 5

that's impossible

Answer 6

why? tan=sin/cos, so if tan=3/4, sin=3

Answer 7

sin is a number between -1 and 1
so sin being 3 is very impossible

Answer 8

hm.

Answer 9

tan(u)=x
is the same as
tan(u)=x/1
draw a right triangle
label one of the angles that isn't 90 deg u
then label the opposite side x
and the adjacent side 1
use Pythagorean theorem to find the hyp

Answer 10

then find sin(u)
you will have enough info to find sin(u) since you will have opposite and hyp

Answer 11

how would you find sin(u) knowing only a 90 degree angle and 1 side?

Answer 12

you know 2 sides

Answer 13

the opp is x
and the adj is 1
find the hyp in terms of x

Answer 14

use Pythagorean theorem

Answer 15

big hint:
opp^2+adj^2=hyp^2

this is Pythagorean theorem

Answer 16

you have opp and adj given to you
just take the square root of both sides

Answer 17

c=sqrt(x^2+1)

Answer 18

hyp is yes sqrt(x^2+1)

Answer 19

so you have opp is x
and hyp is sqrt(x^2+1)

recall sin(u)=opp/hyp

Answer 20

\[\sin(\arctan(x))=\frac{x}{\sqrt{x^2+1}}\]

Answer 21

you have now wrote sin(arctan(x)) as an algebraic expression

Answer 22

you can use that algebraic expression to evaluate sin(arctan(3/4))

Answer 23

where did the x in the numerator come from?

Answer 24

got it.

Answer 25

remember
you were suppose to draw a right triangle for tan(u)=x/1
so I asked you to labeled the opposite side x
and the adjacent side 1
and then I thought you used that to find the hyp

Answer 26

So 3/4=x/sqrt(x+1)

Answer 27

not sure where you got that equation
i thought you wanted to evaluate sin(arctan(3/4))

remember we got sin(arctan(x)) =x/(sqrt(x^2+1))
just replace x with 3/4
to find sin(arctan(3/4))

Answer 28

oh!

Answer 29

\[\sin(\arctan(\frac{3}{4}))=\frac{\frac{3}{4}}{\sqrt{(\frac{3}{4})^2+1}} \\ \]

Answer 30

Is it 3/5 or 12/25? My calculator says the former, my work, the latter...

Answer 31

@freckles

Answer 32

nevermind.