Academy Algebra II/Trig
Pre-Calculus (5.1, 5.2)
8.3: Trigonometric Identities
HW: today: none,
Tomorrow: p.625-626 (20, 24, 32, 38, 50, 54, 78, 86)
Quiz 8.3 (PC 5.1-5.2): Monday, May 19th
(Bio-Testing: 20th and 21st)
Trig Identities
• Reciprocal Identities
1
sin u 
csc u
1
csc u 
sin u
1
cos u 
sec u
1
sec u 
cos u
• Quotient Identities
sin u
tan u 
cos u
cos u
cot u 
sin u
1
tan u 
cot u
1
cot u 
tan u
Trig Identities
• Pythagorean Identities
sin u  cos u  1
2
2
1  tan 2 u  sec 2 u
1  cot u  csc u
2
2
• Even/Odd Identities
sin( u )   sin u csc( u )   csc u
cos( u )  cos u
sec( u )  sec u
tan( u )   tan u cot( u )   cot u
Trig Identities
• Cofunction Identities (section 7.6)


sin   u   cos u
2



cos  u   sin u
2



tan   u   cot u
2



cot   u   tan u
2



sec  u   csc u
2



csc  u   sec u
2

Simplify the Trig Expression
sin x cos x  sin x
2
Simplify the Trig Expression
P.381 (#22)
(1  sin x)(sec x)
2
Check using your calculator:
Y1= Question
Y2= Answer
Check: graphs/tables are
the same
P.381 (#30)
Verify.


cos   x 
2
  sin x tan x
cos x
2
Verify the Identity Algebraically.
P.381 (#46)
cos  sec   cos   sin 
2
2
Verify the Identity Algebraically.
P.381 (#56)
1  csc 
 sec 
cot   cos 
Factor the expression and simplify.
P.381 (#64)
sec x tan x  sec x
2
2
2
5.2: Verifying Trig Identities
1.) Work with one side of the equation.
2.) Look for opportunities to factor an expression,
add fractions, square a binomial, or create a
monomial denominator.
3.) Look for opportunities to use the fundamental
identities.
4.) If the preceding guidelines do not help, try
converting to all sines and cosines.
5.) Try something! Even making an attempt that
leads to a dead end gives insight.
P.381 (#66)
Verify.
csc x  1
 csc x  1
csc x  1
2
Multiply and simplify.
P.381 (#72)
cot x  csc xcot x  csc x
P.381 (#76)
Verify.
1
1
2

 2 cot x
sec x  1 sec x  1
Verify the Identity.
sec   1
2
 sin 
2
sec 
2
Verify the Identity.
1
1
2

 2 sec 
1  sin  1  sin 
tan
Verify the Identity.
2


x  1 cos x  1   tan x
2
2
Verify the Identity.
tan x  cot x  sec x csc x
Verify the Identity.
cos x
sec x  tan x 
1  sin x
Verify the Identity.
2
cot 
1  sin 

1  csc 
sin 
• Homework: Study for Quiz 8.3 tomorrow
– Practice: p.625-626 (25, 33, 35, 37, 39, 41, 45,
55, 61)