Tech Math 2 Notes
Section 20.1
Page 1 of 3
Section 20.1: Fundamental Trigonometric Identities
Big Idea: Expressions containing trig functions can be simplified a lot of times using identities that replace
multiple functions with fewer functions.
Big Skill: You should be able to simplify trigonometric expressions using the identities in this section.
Starting point: For Lab #1, some people got different formulas for s1:
However, both these formulas give the same, correct answers, because for a 6/12 pitch,  = tan-1(6/12)  26.57,
and sin(26.57)  0.4473, and cos(90 - 26.57) = cos(63.43)  0.4473. Thus, we have seen that
sin   cos  90    . This kind of relationship is true in general as follows, and all these relationships are
called “the complementary angle identities”
sin  90     cos
Complementary Angle Identities
tan  90    cot 
sec  90     csc
cos  90     sin 
cot  90    tan 
csc  90     sec
In fact, the “co” part of the cosine, cotangent, and cosecant is short for complementary.
Tech Math 2 Notes
Section 20.1
Page 2 of 3
Next: how come I don’t have buttons on my calculator for cotangent, secant, and cosecant?
Look at the following right triangle and compute side b:
opp
b

 b  sin 20
hyp 1.000
We can also get b using the cosecant function:
hyp 1.000
1
csc 20 

b
opp
b
csc 20
sin 20 
Since these both give the correct formula for b, it must be true that: sin 20 
as: csc 20 
1
.
sin 20
This is an example of one of the reciprocal identities. Since sec 
tan  
1
, which can also be written
csc 20
hyp
adj
adj
and cos 
, and cot  
and
adj
opp
hyp
opp
, we can get two more reciprocal identities:
adj
1
csc 
sin 
Reciprocal Identities
1
sec 
cos 
cot  
1
tan 
Looking at the same right triangle:
opp
sin 20 hyp opp
adj
opp


 tan 20 . This gives us the ratio identities:
and cos 20 
, so
sin 20 
cos 20 adj
adj
hyp
hyp
hyp
Ratio Identities
sin 
cos 
tan  
cot  
cos 
sin 
Tech Math 2 Notes
Section 20.1
Page 3 of 3
Looking at the same right triangle a third time:
The Pythagorean Theorem tells us that: a2 + b2 = c2.
But: b = sin 20, and a = cos 20. Plugging that into the Pythagorean Theorem:
(cos 20)2 + (sin 20)2 = 1
This equation is true for any angle, and leads to the Pythagorean Identities:
Pythagorean Identities
2
2
sin   cos   1
1  tan 2   sec 2 
sin 2   1  cos 2 
tan 2   sec 2   1
1  cot 2   csc 2 
cot 2   csc 2   1
cos 2   1  sin 2 
So, now that we have all these identities, what are they good for?
 They are good for simplifying formulas that have trigonometric functions.
Example:
w
0.5w

 sin  . Show that this simplifies to:
From Lab #1, we got L7  0.5 L tan  
cos  cos 
w
L7  0.5 tan   L  w  
.
cos 
La dee dah…
Tips for Simplifying Trigonometric Expressions or Proving Trigonometric Identities:
 Sometimes it helps to write everything as sines and cosines.
 Look for pieces that can be simplified using the Pythagorean Identities.
 Sometimes it helps to multiply things out.
 Sometimes it helps to factor things.
 Sometimes you’ll have to find common denominators to add fractions.