Trigonometry basic identities

sin( A ) = opp / hypot = a / c	csc( A ) = 1 / sin( A ) = hypot / opp = c / a
cos( A ) = adj / hypot = b / c	sec( A ) = 1 / cos( A ) = hypot / adj = c / b
tan( A ) = sin( A ) / cos( A ) =   opp / adj = a / b	cot( A ) = 1/ tan( A ) = adj / opp = b / a

sin2(x) + cos2(x) = 1	tan2(x) + 1 = sec2(x)	cot2(x) + 1 = csc2(x)
sin(x y) = sin x cos y cos x sin y
cos(x y) = cos x cos y sin x sin y

tan(x y) = (tan x tan y) / (1 tan x tan y)

sin(2x) = 2 sin x cos x

cos(2x) = cos²(x) - sin²(x) = 2 cos²(x) - 1 = 1 - 2 sin²(x)

tan(2x) = 2 tan(x) / (1 - tan²(x))

sin²(x) = 1/2 - 1/2 cos(2x)

cos²(x) = 1/2 + 1/2 cos(2x)

sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )

cos x - cos y = -2 sin( (x-y)/2 ) sin( (x + y)/2 )

Trig Table of Common Angles

angle	0	30	45	60	90
sin2(a)	0/4	1/4	2/4	3/4	4/4
cos2(a)	4/4	3/4	2/4	1/4	0/4
tan2(a)	0/4	1/3	2/2	3/1	4/0

Note: The above are "squared" values of sine, cosine and tangent. The "square root" of these must be used for normal sine, cosine and tangent applications.

a / sin(A)  =  b / sin(B)  =  c / sin(C)       (Law of Sines)

LAW OF SINES: The ratio of any side of a general triangle when divided by the sine value of the angle opposite the side is equal to the ratio of any other side of a general triangle when divided by the sine value of the angle opposite it (the other side).

c2  =  a2  +  b2  -  2ab cos(C) b2  =  a2  +  c2  -  2ac cos(B) a2  =  b2  +  c2  -  2bc cos(A)

(Law of Cosines)

LAW OF COSINES:  The square of any side of a general triangle is equal to the sum of the squares of the other two sides decreased by (minus) two times the multiplication product of the (two not included) sides multiplied times the cosine of the angle between the two (not included) sides.

(a - b)/(a + b) = tan 1/2(A-B) / tan 1/2(A+B) (Law of Tangents) -- not necessary with the above

Trig Values of Angles Greater Than 90^o

Dealing with angles great than 90 degrees is a relatively easy task. Basically, you construct or place the angle in a rectangular coordinate system ... with the vertex of the angle at the origin of the coordinate system, the initial starting side of the angle along the positive "x" axis of the coordinate system, and the terminal side of the angle rotated counter clockwise (from the initial side) to wherever it may fall (second, third, fourth quadrant). Technically, you then select a point on the terminal side ... drop a perpendicular to the horizontal axis ... forming a small right triangle. The trig values of the angle at the origin of this new triangle are defined as being the trig values of the angle you are working with.

Since this is complicated (at least in description) ... a shortcut is usually used to find trig values of angles greater than 90 degrees. Find the quadrant in which the terminal side of the angle you are dealing with falls. Subtract 180 or subtract from 180 or subtract from 360 ... per the picture above. The trig value of the angle you are dealing with will be the same as the trig value of the answer you get. Finally, depending upon the particular trig value you are finding and the quadrant the angle is in, make your answer positive or negative per the diagram above.

Example:  Find the cosine value of a 150 degree angle. Step 1: 150 degrees falls into the second quadrant. Step 2: Take 180 -150 (per diagram above) ... yielding 30. {Conclusion: The numeric cosine value of 150 is the same as the cosine value of 30.}Finally, attach the correct positive/negative sign. In the second quadrant ... cosine values are negative (per diagram above). Therefore: the cosine 150 = - cos 30. Since the cosine of 30 is sqrt(3) / 2 ... the cosine of 150 is - sqrt(3) / 2.