Question

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

Answer 1

Have you learned SOH CAH TOA

Answer 2

they are teh same

Answer 3

Yes I know what SOH CAH TOA is.

Answer 4

so sin (x) is going to be the side opposite / hypotenuse

what is the side opposite of x?
and what is the length of the hypotenuse?

Answer 5

The triangle is a right angle triangle and you should be able to calculate the hypotenuse
(if you don't already notice that it's a 3-4-5 right angle triangle)

Answer 6

So how exactly do you calculate the hypotenuse?

Answer 7

We can use Pythagorean theorem

a^2 + b^2 = c^2

a and b are the two legs or bases of the triangle, and c is the hypotenuse
the hypotenuse is the side opposite the 90 degree angle

Answer 8

so our two sides of the triangle are 3 and 4 so

3^2 + 4^2 = c^2

can you solve for c?

Answer 9

So 5?

Answer 10

Hello..?

Answer 11

Yes

Answer 12

So our hypotenuse is 5

what is the side opposite of x

Answer 13

3..?

Answer 14

Perfect!!

Answer 15

So 3/5?

Answer 16

yessirrrr he got it right

Answer 17

so sin (x) = opposite/ hypotenuse

and we said

Answer 18

Exactly!!

Answer 19

which is 0.6

Answer 20

Right?

Answer 21

Now let's do cos (y)

cos (y) = adjacent/ hypotenuse

we already know the hypotenuse is 5

but now we're looking for the side adjacent to the angle y

Answer 22

Yes, 3/5 can be simplified to 0.6

Answer 23

But isn't 3 the adjacent side?

Answer 24

It is!

For angle x, the side OPPOSITE to it is 3

For angle y, the side ADJACENT to it is 3

Answer 25

So what would cos (y) be

Answer 26

The same, 3/5. Simplified to 0.6

Answer 27

Bingo!

Answer 28

So what we learn here is that they are equal to each other

and we get a trigonometric identity from this
sin (x) = cos(90 - x)
cos (x) = sin(90 -x)

Answer 29

Can you explain to me trigonometric identity?

Answer 30

Trigonometric identities are basically formulas in trigonometry that are always true

so for example, remember Pythagorean theorem? The a^2 + b^2 = c^2 for a right angle triangle

we see that if we do sin^2 (x) + cos^2 (x) = 1

Answer 31

how much you're expected to know those formulas really depends on the class you're taking, but I wouldn't worry too much about it if it's just a regular geometry class

Answer 32

Ok, so do I use that for the relationship between sin x and cos y?

Answer 33

To answer your question, you can say that sin (x) and cos (y) are equivalent!

If you want to use the sin(x) = cos(90 - x) that's totally up to you!
But you can see that this formula is true because in our right angle triangle,

we have three angles and they should all add up to 180
but since it's a right angle triangle, we know that one angle is 90 degrees
that means the other two angles add up to 90

and that's why sin (x) = cos (y)
because 90 - x would give us the value of y

so this formula of sin(x) = cos(90 - x) is true

Answer 34

hopefully that makes sense :)

Answer 35

So would it be satisfactory if you only say they're equivalent?

Answer 36

If you haven't learned about that formula and whatnot, I'd say absolutely! But since I'm not your teacher, I can't exactly guarantee what would be a satisfactory answer haha

Answer 37

But I can tell you that your final answer is correct :)

Answer 38

Ok so if you have time, could you help me with another one?

Answer 39

Sure! Tag me in your new post like @AZ

Answer 40

Oh ok!