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- Unofficialllyy

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- Unofficialllyy

How do similar right triangles lead to the definitions of the trigonometric ratios?

- Unofficialllyy

@jim_thompson5910 Could you try to help?

- Unofficialllyy

My notes say this:
The term trigonometry comes from a Greek word meaning “triangle measuring.”
The sine of an angle within a right triangle is found by dividing the length of the opposite side by the length of the hypotenuse.
The cosine of an angle within a right triangle is found by dividing the length of the adjacent side by the length of the hypotenuse.
The tangent of an angle within a right triangle is found by dividing the length of the opposite side by the length of the adjacent side.
But I am not sure which one is the right answer for the question.

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## More answers

- jim_thompson5910

so the question is `How do similar right triangles lead to the definitions of the trigonometric ratios?`
and there are no answer choices? it's a fill in the blank kind of question?

- Unofficialllyy

This is one of the questions I am going to be asked on my oral test, so I want to make sure I know how to answer it. There are no multiple choice answers.

- jim_thompson5910

let's say we had this right triangle
|dw:1433986107369:dw|

- Unofficialllyy

Alright

- jim_thompson5910

cut each length in half to get this similar triangle
|dw:1433986215952:dw|

- jim_thompson5910

oops typo
|dw:1433986258148:dw|

- jim_thompson5910

let's say we place the reference angle theta here
|dw:1433986304712:dw|

- jim_thompson5910

for the large triangle, we know
opposite/hypotenuse = 6/10 = 3/5 = 0.6
for the small triangle, we know
opposite/hypotenuse = 3/5 = 0.6

- jim_thompson5910

the ratio of those two sides forms the sine of the angle theta

- jim_thompson5910

so if we want to make a similar triangle, we just need to make sure that the ratio of 0.6 stays the same

- jim_thompson5910

|dw:1433986433430:dw|

- Unofficialllyy

I just don't understand the question. It should be on the notes page of my lesson but I can't seem to find it.

- Unofficialllyy

Do they relate because the sides are always proportional and then the sine, cosine, and tangent is found by dividing the lengths?

- jim_thompson5910

you agree that similar right triangles have the same angles right?

- Unofficialllyy

Yes

- jim_thompson5910

so let's say you didn't have a protractor and you couldn't measure the angles
how can you check to see if the two angles are congruent?

- Unofficialllyy

Ummm using AAA or the other ones lol

- jim_thompson5910

but we know nothing about the angles

- jim_thompson5910

let's say you had only a ruler

- jim_thompson5910

using the ruler, you can measure the segments but not the angles

- Unofficialllyy

Right

- jim_thompson5910

you can use the converse of the SSS similarity theorem to say
if the sides are in proportion, then the triangles are similar
which will lead you to concluding the angles are congruent

- jim_thompson5910

with right triangles, you only need to worry about 2 sides (not all 3)
and the trig functions are simply ratios of the sides

- Unofficialllyy

So how do similar right triangles and the trig ratios relate? I don't get it

- jim_thompson5910

For similar right triangles, the sine ratio will stay the same. The cosine ratio will stay the same. The tangent ratio will stay the same

- jim_thompson5910

Again the trig ratio is used to connect the angle with two sides
the angle will stay the same since we're dealing with similar triangles (with congruent angles)

- Unofficialllyy

Oh so the trigonometric ratios will always be the same with similar right triangles

- jim_thompson5910

|dw:1433987739690:dw|

- jim_thompson5910

as long as theta stays where it is, for any other similar triangle you can think of, the sine ratio will stay put at 0.6

- jim_thompson5910

the sides may change, but their relationship or ratio will stay the same

- jim_thompson5910

for example, the opposite side may turn into 600, but the hypotenuse will also change to 1000
600/1000 = 0.6
the sine ratio stays the same

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