As light passes from one substance, such as air, to another substance, such as glass, the light is bent.  The relationship between the angle of incidence (∠

.  This phenomenon, now a law of Physics and Mathematics, indicates how trigonometric ratios can allow an interested party to discern information using observable principles.

Right Triangles have many useful applications based on the built-in requirements and proportions they have.  If you’ll recall from Geometry class, a right triangle has - you guessed it - three interior angles, one of which 

be a 90° angle, and thus having two others that 

add up to the remaining 90°.  This difference in angle measurement for the “other” two angles is always in a proportion, which you can use to define almost all aspects of trigonometry.

The two sides that make up the right angle (seen above with the ⬜ symbol) are called the 

of the triangle, and the third, longest side is called the 

.  The two angles besides the ⬜ are acute in measurement, made up of one leg and the hypotenuse.  If a leg is a part of one of the acute angles, it is known as the 

 side to that angle, and the opposite leg is known as the 

side.  These terms describing sides are an integral part of filling in the trigonometric ratios that we will use going forward.

Also to be mentioned is the relationship laid out by the three sides, made famous by the Pythagorean Theorem.  Also pictured above, the lengths of the legs are assigned the vairables 

2.  The legs given have the lengths 8 and 6, and the hypotenuse is found by adding their squares, then taking the square root of the answer.

Can you prove the Pythagorean Theorem by using a quadrilateral as an example?  Write your explanation in your journal.

After that trip, let’s take a look at what the angles tell us.  If the two acute angles of a right triangle are congruent in measurement to another triangle, these triangles are 

.  When looking at one of these congruent angles in the similar triangles, the ratio of the corresponding sides are equal.  Therefore, any two congruent angles of different right triangles will have equal ratios associated with them.  The ratios of the sides of the right triangles can be used to define 

Consider angle θ (∠θ).  The ratio of the side opposite of θ and the hypotenuse is known as the 

.  The ratio of the side adjacent (part) of θ and the hypotenuse is called the 

.  The ratio of the side opposite θ and the side adjacent to θ is called the 

.  These relationships can be remembered mnemonically as SOHCAHTOA.

list the sine, cosine and tangent ratios and their values for ∠θ in your Math journal.

Given:  ΔABC.   Side AC has a length of 18m.  Side AB has a length of 33m and is the hypotenuse.  Find the length of side BC.  Then, find the sine, cosine and tangent from ∠B - answers can contain a square root sign 

The reciprocal of any ratio is also true for trigonometric ratios.  These reciprocals are equally valued as SOHCAHTOA, just the inverses.  The first, 

, is equal to 1 / sin θ, or hypotenuse over the length of the opposite side.  

 is the name for 1 / cos θ, or hypotenuse over the adjacent side and 

 is 1 / tan θ, or the adjacent side over the opposite side.  This relationship is cyclical, as the SOHCAHTOA ratios are the reciprocals of these, and so on.

Given: ΔMNP.  Leg MP has a length of 10m and leg MN has a length of 7m.  Find side NP.  Then, find the sine, cosine, cosecant, secant, cotangent and tangent from ∠P.  Answers can contain a square root sign 

In right triangles, there are certain relationships of more regular angles: 45°, 45°, 90° and 30°, 60°, 90°.  These triangles have the same ratios of sides thanks to trigonometric ratios.

Notice that, in a 30°-60°-90° triangle that sin 30 = cos 60, and that cos 30 = sin 60.  We use these examples to relate this similarity in ratio to all triangles, called 

Students will dive deeper into right triangles and the unique properties they have to provide.

Think Rationally, Trigonometrically