Download Section 5.1 * Using Fundamental Identities

Section 5.1 – Using Fundamental Identities
Honors
PreCalculus
In this section you will learn how to use Fundamental Trig Identities to evaluate trigonometric functions, simplify
trigonometric expressions, and rewrite trigonometric expressions.
FUNDAMENTAL TRIGONOMETRIC IDENTITIES
Reciprocal Identities
sin u = ________
cos u = ________
tan u = ________ csc u = ________
sec u = _______ cot u = ________
Quotient Identities
Pythagorean Identities
Tan u = ________ cot u =________
sin2u + cos2 u= ________ 1 + tan2u = ________ 1 + cot2u = ________
Cofunction Identities
Sin (

- u ) = _______
2
cos (

- u ) = _______
2
tan (

- u ) = ________
2
cot (

- u ) = ________
2
sec (

- u ) = ________
2
csc (

- u ) = ________
2
Even/Odd Identities
Sin (-u) = ________
cos (-u) = ________
tan (-u) = ________
csc (-u) = ________
sec (-u) = ________
cot (-u) = ________
Use the given values to evaluate (if possible) all six trigonometric functions.
1.
csc  
25
7
, tan  
7
24
3. tan  is undefined, sin  > 0
2. sec x = 4, sin x > 0
4. csc   5, cos   0
Section 5.1 – Using Fundamental Identities
Honors
PreCalculus
Use the fundamental identitie to simplify the expression.
5. sec2 x(1  sin 2 x)
8.
sin  csc 
tan 
11. csc  tan   sec 
1
tan 2  1
6.
csc x
sec x
7.
9.
tan 2 
sec 2 
10. cos t (1  tan 2 t )
12. sin 2 x csc 2 x  sin 2 x
13. cos 2 x  cos 2 x tan 2 x
16. sec4 x  tan 4 x
14.
cos 2 x  4
cos x  2
15. 1  2 cos 2 x  cos 4 x
17.
1
1

sec x  1 sec x  1
18.
tan x
1  sec x

1  sec x
tan x
19. tan x 
sec2 x
tan x
Section 5.1 – Using Fundamental Identities
Honors
PreCalculus
Use Trigonometric substitution to write the algebraic expression as a trigonometric function of
 where 0   

2
20.
64  16 x 2 , x  2cos 
21.
49  x 2 , x  7sin 
22.
x 2  4, x  2sec
23.
x 2  100, x  10 tan 
24.
9x2  253x  5tan 
25.
10  x 2 , x  10 sin 
26. Use trigonometric substitution u  a tan  ,where 

2
 

2
and a  0 , to simplify the expression
a2  u
.