Question

@phi
The moon forms a right triangle with the Earth and the Sun during one of its phases, as shown below: (A triangle with sides Earth, Sun, and Moon. Side Earth-Sun has distance y. Angle x is on the same side) A scientist measures the angle x and the distance y between the Earth and the Sun. Using complete sentences, explain how the scientist can use only these two measurements to calculate the distance between the moon and the Sun.

Answer 1

can you post or draw the picture?

Answer 2

if we call the points M,E and S
where is the right angle? which side is the hypotenuse ?

Answer 3

Sorry, computer slow.

Answer 4

Sorry, computer slow.

Answer 5

this looks like trig.

you know angle x and side y
you want to know the side opposite angle x

do you know SOH CAH TOA ?

Answer 6

Sorry, computer slow.

Answer 7

yes, its just that when the angle is on is on the same side as Y it is throwing me off on which function to use.

Answer 8

do you know how to "name" the sides?
which side is the hypotenuse, and standing at angle x, which side is
1) adjacent
2) opposite
?

Answer 9

ME is the opposite
ES is the adjacent
MS is the hypotenuse

Answer 10

they say they "measured side y" which is ES = adj
you want to find ME= opp
so I would use the trig function that uses opposite and adjacent

Answer 11

Actually it wants me to find MS if I am not mistaken, says to use given information (x and y) to find the distance between the moon and the sun.

Answer 12

good point. MS is hypotenuse
and you know ES = adj

so use the trig function that uses hyp and adj

Answer 13

Cosine or Secant.

Answer 14

nobody uses secant these days, except for specialized reasons

Answer 15

Okay, so I would start of with
Cos = Adj/Hyp
Cos x = y/x

Answer 16

x is used for the angle, so I would use a different letter for the unknown side.
how about d (for distance) ?

Answer 17

so y/d?

Answer 18

as long as you know d= MS
cos x = y/d
solve for d

Answer 19

so,
Cos = Adj/Hyp
Cos x = y/d
x = d

Answer 20

Cos x = y/d

to solve for d, you can multiply both sides by d
d cos x = y*d/d

the right side simplifies to y*1 or just y
d cos x = y

now divide both sides by cosx
d = y/cos x

Answer 21

So how would this look as a whole? Just making sure I getting it.

Answer 22

try writing up what you think the answer is

Answer 23

Cos = Adj/Hyp
Cos x = y/d (or y*d/d, having trouble with this part)
Cos x = y
d = y/cos x

Answer 24

or Cos = Adj/Hyp
Cos x = y/d
x (x) = d
x= d

Answer 25

start by saying
The given right triangle with angle x, has measured side y (adjacent side),
and unknown side d (side MS, or hypotenuse)
thus we should use CAH,
Cos x= Adj/Hyp
Cos x = y/d

the next step is algebra
\[ \cos x = \frac{y}{d} \]
if two sides are equal, and we multiply both sides by the same thing, they stay equal
in other words, if we multiply both sides by "d", we still have an equation
\[ d \cdot \cos x = d \cdot \frac{y}{d} \]

however \( d \cdot \frac{y}{d} \) is the same thing as \( \frac{d \cdot y}{y} \) and that is the same thing as \( y \cdot \frac{d}{d} \) and that is the same as \( y \cdot 1\) or just y

Answer 26

so
\[ d \cos x = y \]
now we want d by itself. Divide both sides by cos x
\[ d \frac{\cos x}{\cos x} = \frac{y}{\cos x} \]
and again, something divided by itself is 1, so
\[ d= \frac{y}{\cos x} \]

Answer 27

thanks your help! can you check my answers to other problem (just three)