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How do similar right triangles lead to the definitions of the trigonometric ratios?
@jim_thompson5910 Could you try to help?
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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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?
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.
let's say we had this right triangle |dw:1433986107369:dw|
Alright
cut each length in half to get this similar triangle |dw:1433986215952:dw|
oops typo |dw:1433986258148:dw|
let's say we place the reference angle theta here |dw:1433986304712:dw|
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
the ratio of those two sides forms the sine of the angle theta
so if we want to make a similar triangle, we just need to make sure that the ratio of 0.6 stays the same
|dw:1433986433430:dw|
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.
Do they relate because the sides are always proportional and then the sine, cosine, and tangent is found by dividing the lengths?
you agree that similar right triangles have the same angles right?
Yes
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?
Ummm using AAA or the other ones lol
but we know nothing about the angles
let's say you had only a ruler
using the ruler, you can measure the segments but not the angles
Right
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
with right triangles, you only need to worry about 2 sides (not all 3) and the trig functions are simply ratios of the sides
So how do similar right triangles and the trig ratios relate? I don't get it
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
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)
Oh so the trigonometric ratios will always be the same with similar right triangles
|dw:1433987739690:dw|
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
the sides may change, but their relationship or ratio will stay the same
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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