Basic Trig

So, recall from class that we have 3 trig ratios:

sin (A) = opp/hyp

cos (A) = adj/hyp

tan (A) = opp/adj

(The reference angle is A in the picture below.)

This is most easily remembered as:

SOH CAH TOA

Or, 

sin (A) = opposite / hypotenuse

cos (A) = adjacent / hypotenuse

tan (A) = opposite / adjacent

So that for a 3-4-5 triangle, with a reference angle A between 4 and 5 (as above):

sin (A) = 3/5 = 0.6

cos (A) = 4/5 = 0.8

tan (A) = 3/4 = 0.75

You can use these sine, cosine, or tangent values to find the angle associated with them.  In the old days, you would look up the angle in a table - the angle that corresponds to a particular sine value, etc.  However, it is much easier to have your calculator do the work for you.  Simply use the 2ND key and either SIN, COS, or TAN, depending on which ratio you wish to use.

2ND SIN 0.6 = 36.7

Or,

2ND COS 0.8 = 36.7

Or, 

2ND TAN 0.75 = 36.7

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It is probably more useful to use the trig functions to find unknown sides in a triangle.

So, imagine that you have a right triangle with other angles 30 and 60, and a hypotenuse of 12.  Let the reference angle be the 30 degree angle  You can solve for the OPP side using SIN, or the ADJ side using COS.  Let's do both.

cos 30 = ADJ / 12

0.866 = ADJ / 12

ADJ = 10.4

And for the other (OPP) side:

sin 30 = OPP / 12

0.5 = OPP / 12

OPP = 6

The beauty of all this is that we don't have to know all of the information in a triangle at the onset - we can calculate what we don't know, as long as we have angles and a side in between.  We use that powerful idea to find the distance to things we can't actually reach:  the Moon, the Sun, nearby stars, or local stuff like heights of buildings and widths of rivers.  It's awesomely powerful!

Sample problems:

1.  Find the 3 trig ratios associated with a 20-21-29 triangle.  Let the angle between 20 and 29 be the reference angle.  After you write down the trig ratios, find the angles.

2.  Repeat for a 95-168-193 triangle.

3.  You are standing 20-m from the base of a tree.  When you look to the top of it, your eyes make a 35 degree angle with the ground.  How tall is the tree?

4.  Consider an 8-15-17 right triangle.  If the hypotenuse is 10 cm long, how long are the other two sides?