Question

Use the image below to answer the following question. What relationship do the ratios of sin x° and cos y° share?

Answer 1

The ratios are both identical (3 over 5 and 3 over 5).
The ratios are opposites (negative 3 over 5 and 3 over 5).
The ratios are reciprocals (3 over 5 and 5 over 3).
The ratios are both negative (negative 5 over 3 and negative 3 over 5).

Answer 2

ok, what kind of triangle is this?

Answer 3

A right triangle, meaning it equals 180 degrees i believe

Answer 4

yes. and the right angle, how many degrees is that?

Answer 5

well, all triangles have angles summing up to 180 degrees. but yes, it is a right triangle.

Answer 6

hello?

Answer 7

I'm still here

Answer 8

Im not sure how many degrees it is

Answer 9

a right angle is always 90 degrees.

Answer 10

Oh yeah thats right! I remember that I think

Answer 11

so if the triangle has a total of 180 degrees, and the right angle is 90 degrees, what is left for \(x\) and \(y\) combined?

Answer 12

90

Answer 13

right. now a pair of angles which add to 90 degrees has a name, do you know what it is?

Answer 14

Hmm I'm not sure, sorry

Answer 15

pair of angles which add to 90 degrees is called a pair of complementary angles
pair of angles which add to 180 degrees is called a pair of supplementary angles

which do we have here?

Answer 16

Complementary since they both are 90

Answer 17

But they also both add up to 180 so I want to say supplementary

Answer 18

no, \(x\) and \(y\) are complementary. no pair of angles here adds to 180.

one thing about sin and cos is that the sin of one angle is equal to the cos of the complementary angle and vice versa.

we can see this if we remember SOHCAHTOA:

Sin = Opposite over Hypotenuse
Cos = Adjacent over Hypotenuse
Tan - Opposite over Adjacent

look at the triangle here. what is the length of the hypotenuse?

Answer 19

5

Answer 20

good. so if angle \(x\) is the angle we are looking at, what is the length of the opposite side?

Answer 21

The opposite would be 3

Answer 22

yes, so how would you express \[\sin x^\circ=\] from what I have told you?

Answer 23

Divide the opposite by the hypotonuse and get the sin?

Answer 24

yes, but what will that give you? give me the actual numbers...

Answer 25

ok, I just had to see if that was the right format to get the answer, one second im calculating it

Answer 26

I got 0.010

Answer 27

just write the fraction

Answer 28

As a fraction it gives me 1,100

Answer 29

sin is opposite over hypotenuse. what is the opposite ? what is the hypotenuse?

opposite / hypotenuse =

Answer 30

0.6

Answer 31

write the fraction!

Answer 32

always write the fraction unless you are told you need a decimal

Answer 33

3/5

Answer 34

Oh okay, good advice

Answer 35

much better.

now, find the cos of angle y. remember, cos is adjacent over hypotenuse

Answer 36

4/5 is what I got when dividing, is it right so far?

Answer 37

are you sure? I asked for cos of angle \(y\). what is adjacent for angle \(y\)?

Answer 38

The adjacent is 4

Answer 39

angle y, not x.

Answer 40

Im not sure

Answer 41

look at the picture. see angle y? it is made up from two triangle sides. what are their lengths? on the picture I am looking at, the side of length 4 does not go into the corner labeled \(y\) so it cannot be the adjacent side, can it?

Answer 42

Ohhh I see what you mean, no it cant

Answer 43

so, this time I am sure you will get it right. what is the fraction for \(\cos y\)?

Answer 44

Im nervous on this one... 3/5?

Answer 45

3/5 is correct. now where have we seen 3/5 before?

Answer 46

Wasnt that the sin for X?

Answer 47

it was!

how about the sin of angle y, what is that? then find the cos of angle x...

Answer 48

How do I find the cos of a fraction?

Answer 49

do the same thing you did before. sin of angle y is going to be opposite of angle y divided by hypotenuse

cos of angle x is opposite of angle x divided by hypotenuse

Answer 50

sin and cos are just ratios of the lengths of the sides

Answer 51

I will be back when you reply

Answer 52

Hmm i think i get it, -3/5?

Answer 53

Nvm, its 3/5 again. Giving me the answer of A because they both are identicle

Answer 54

Thank you so much! I appreciate your patience with me

Answer 55

yes, if you have complementary angles, the sin of one is always equal to the cos of its complementary angle, and vice versa