YOU are better than YOU think. Show yourself  how:

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 Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and writing

Do not leave here without it -  Logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

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After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

Explore collaborative whiteboards from groupboard,  twiddla  or scriblink.

Similarity of Right Triangles

Two right triangles are similar if they have an angle in common BESIDES a right angle. More generally, two triangles are similar if and only the angles in one equal the angles of the other in some order. But that topic is not of interest today. 

For similar triangles, the ratio of their matching or corresponding sides are equal.  The diagram below illustrates this in a special case.

The observation or assumption that

the ratios of side lengths, here adjacent/hypotenuse or opposite/hypotenuse are independent of the scale factor and depend only the angle the hypotenuse makes with the adjacent side

justifies the definition and calculation of trig functions using the ratio of sides for similar right triangles.  See below.

Trigonometric Functions of Acute Angles

Given a right triangle with acute angle , say 

we may form the ratios

The following chains of reason show that these ratios depend only on the angle and are independent of the scale or size of the triangle we use to compute the ratios.

Why computation does not depend on scale.

Suppose we have two right triangles

with a common angle as shown.  

By the AA minimal condition for similarity of triangles, the presence of the common angle  in addition to the common right angles in each triangle implies the two triangles are similar. Therefore, there is a proportionality constant K such that

a = Kd, b= Ke and c = Kf.

Therefore the ratio of adjacent over hypotenuse 

coincides in each triangle. So this ratio does not depend on the right triangle from which it comes provided there is a common angle .

Like wise the ratios of opposite over hypotenuse

and opposite over adjacent

do not depend on the right triangle from which they comes provided there is a common angle .