Why are the trigonometric ratios not affected by the size of the right triangle?
Answers
Answer:
Because for a given angle (argument of the trigonometric function), all right triangles having that given angle are similar triangles (all three angles have equal measure). The triangles have the same “shape,” so that the ratios of corresponding sides (opposite the same considered angles) are equal.
There are only three interior angles, and the trigonometric functions (using this particular definition) are given in terms of some angle. If you’ve fixed two of the three angles (one required to be a right angle, the other the argument of the trigonometric function you are considering), then the remaining angle is also fixed, since the sum of the three angles is fixed to a constant value (180 degrees or pi radians). If two triangles have the same three angles, they are similar, and the ratios of corresponding sides (the ratios defining the trigonometric fuctions) are also equal, regardless of the size of the triangles.
Answer:
when you increase the size of a triangle, without changing the angles, the side lengths all increase at the same rate relative to one another.
Step-by-step explanation:
Consider the triangles above:
There are two smaller triangles, A and B , inside a bigger one, C . If you look closely, you can see that C has the same angles as both A and B . But, are the ratios of the side lengths the same? Let’s find out!
Let’s say the side lengths of B are BG, BV, and BH (BG is ground length, BV is vertical length, and BH is hypotenuse), and similarly, we have AG, AV, AH, and CG, CV, CH. How do write the lengths for triangle C in terms of A and B ? It should be clear:
Hypotenuse (H): This is the easiest one. It’s quite clear that BH + AH = CH.
Ground (G): Use the rectangle! We know it’s a rectangle because the upper left corner of the rectangle, Φ, must be added to theta and 90 - θ to make 180°, because CH is a line, and a line is 180°. The third angle is 90 - θ because the sum of all angles of a triangle is 180°, and if you look at B, you’ll see that 90 + θ + Φ is 180°, so Φ has to be 90 - θ. So, anyway, if we solve for Φ, we get Φ = 90°, a right angle! I hope it’s clear that we’re dealing with a rectangle now. One property of rectangles is, all parallel components have the same length. So, that means CG = AG + BG.
Vertical (V): Same argument as Ground! CV = AV + BV.
As you can see, triangle C is just the “growth” from triangle B to triangle A . By just adding the side lengths of triangle A to triangle B , you get another similar triangle, C .
Let’s verify that the ratios are, in fact, consistent:
CVCG=AV+BVAG+BG . If you multiply CG by some number, call it λ, you will get CV. The problem is proving that doing the same to BG and AG will give you BV and AV respectively. What I can tell you is, if you do make this assumption, you will indeed be able to see that AV+BVAG+BG=λAG+λBGAG+BG=λ(AG+BGAG+BG)=λ . This checks out, and I claim that any other assumption will not check out due to how “linear” this equation is: other assumptions will yield different results due to the uniqueness of this equation.
