This document summarizes fundamental trigonometry concepts needed for calculus, including: 1) Definitions of the six trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) in terms of ratios of sides of a right triangle. 2) Reciprocal trigonometric functions and cofunctions. 3) Signs of the trig functions in the four quadrants. 4) Converting angles greater than 90 degrees to less than 90 degrees using quadrant rules. 5) Fundamental trigonometric relationships including Pythagorean identity and even-odd properties. 6) Conversion between degrees and radians. 7) Some basic trigonometric values.

1. 00 30 /6 45/4 60 /3 90 /2
sin 0 1/2 2 /2 3/2 1
cos 1 3/2 2 /2 1/2 0
tan 0 3/3 1 3 --
cot -- 3 1 3/3 0
sec 1 2 3/3 2 2 --
csc -- 2 2 2 3/3 1
Fundamental Trigonometry Concepts Needed for Calculus
sin A =
hyp
opp
r
y
c
a

cos A =
hyp
adj
r
x
c
b

tan A =
adj
opp
x
y
b
a
 cot A =
opp
adj
y
x
a
b
 sec A =
adj
hyp
x
r
b
c
 csc A =
opp
hyp
y
r
a
c

1. Reciprocal Functions: csc A =
Asin
1
; cot A =
Atan
1
; sec A =
Acos
1
2. Cofuntions: sin & cos, tan & cot, sec & csc. Any function of an angle less than 90 is equal to the
cofunction of its complement (i.e. 90 - A).
3. Signs on the trig functions I the four quadrants are:
All trig functions are positive in Q I. cos and sec are negative in Q II and Q III
sin & csc are negative in Q III & Q IV tan and cot are negative in Q II and Q IV
4. To take the function of an angle A greater than 90 and reduce it to the same function of an angle less
than 90 first determine the sign on that function in that particular quadrant and then:
i) if A is in Q II, use 180 - A; ii) if A is in Q III, use A - 180; iii) if A is in Q IV use 360 - A.
5. FUNDAMENTAL RELATIONSHIPS:
sin (-) = - sin ; cos (-) = cos ; sin2
 + cos2
 = 1 for all .
 sincoscossin)sin(   sinsincoscos)cos( 
6. Other relationships that follow from (5) above are: tan2
 + 1 = sec2
 & 1 + cot2
 = csc2

cos 2 = cos2
 - sin2
; sin 2 = 2 sin  cos ;
2
2cos1
cos2 


 ;
2
2cos1
sin2 



7.Relationship between degrees and radians (a pure number, no units): o
DR
180


; R-radians; D-degrees
8. Basic trig values: or
sin 0 = 0 /2 = 0.000 = cos 90
sin /6 = 1 /2 = 0.500 = cos 60
sin /4 = 2 /2 = 0.707 = cos 45
sin /3 = 3 /2 = 0.866 = cos 30
sin /2 = 4 /2 = 1.000 = cos 0