What geometric principal allows us to define trigonometric functions as ratios of sides of triangles and be confident that they are indeed functions? that is, how do we know that the value of each angle put into a trigonometric function results in exactly one output value?

Solution:

The  geometric principal which allows us to define trigonometric functions as ratios of sides of triangles and they are indeed functions

1. Similarity of Two Right Triangles

2. Concept of unit circle

→→As a complete circle means angle of π radians or an angle of 360°.So if you take any of trigonometric function and any right angled triangle , you will find that for angle between 0°≤Ф≤90°, values of trigonometric functions are different.Because for each angle, ratio of side lengths of right angled triangle changes, that's why for each unique angle we get a unique output value for different trigonometric functions.

You can call it  one-one onto mapping (function) also.

answ9x x 3=

step-by-step explanation:

d

step-by-step explanation:

hello and welcome!

so, when we see the words "proportional" along with a table, we need to know that we're looking numbers that divide by the same thing.

looking at this table, we simply have to do a trial and error, and divide to see which numbers have the same quotient. if you look at choice a & c, they both have equal quotients for three of the numbers, with one exception, so those can't be the answer. looking at b, there is no similarity, so that can't be it. then, we see d; if you divide all the numbers in the table, you'll see that they all divide by two, which is a proportional relationship, and our answer!

also, always remember that the x will always be the number that goes into y, and somewhere behind this table, is a formula. for this specific table, the formula would be:

since our x-value is being divided by two to get the y-value. and if they were to ask for more numbers in the series, this is the formula you would use to find them.