Download Precalculus Pre-AP Name__________________________

Precalculus Pre-AP
Identities Investigation
Name__________________________
The following ratios are given.
sin( ) 
y
r
cos( ) 
x
r
tan( ) 
y
x
csc( ) 
r
y
sec( ) 
r
x
cot( ) 
x
y
Use the ratios above to verify the following “reciprocal identities”.
sin( ) 
1
csc( )
cos( ) 
1
sec( )
tan( ) 
There are 3 other “reciprocal identities”. Name them here:
Use the ratios above to verify the following “quotient identities”.
tan( ) 
sin( )
cos( )
cot( ) 
cos( )
sin( )
1
cot( )
Use the ratios from the previous page and the fact that x2 + y2 = r2 to verify the following “pythagorean identities”.
sin 2 ( )  cos 2 ( )  1
1  cot 2 ( )  csc2 ( )
1 tan 2 ( )  sec2 ( )
Summarize the fundamental trigonometric identities: (6 reciprocal, 2 quotient, and 3 pythagorean)
Reciprocal Identities
Quotient Identities
Pythagorean Identities
These fundamental identities can be used to simplify trigonometric expressions and to verify other identities.
Simplify the following expressions
cot(t )
csc(t )
csc(t)tan(t)
cos 2 (t) (sec2 (t)-1)
tan2(t )cos2(t) + cot2(t)sin2(t)
sec2(t ) - 1
1 – sin2(t )
sin(t)csc(t)-cos2(t)
:
csc2(t ) - cot2(t )
Don’t use a calculator to simplify these.
tan(20°)-
sin( 20)
cos( 20)
sin(80°)csc(80°)
Factor the following. Then simplify if possible
sin2(t ) – 4sin(t ) - 21
sin 2(x ) - cos 2(x ) sin 2(x )
tan(10°)cot(10°)
Before attempting to simplify the next few, review how to deal with fractions algebraically:
a b
Write  as a single fraction.
x y
ab
Write
as a sum of 2 fractions.
z
Simplify the following expressions.
sin( t )
cos(t )

1  cos(t ) sin( t )
2  sin( t )
cos(t )
1
1

1  cos( x) 1  cos( x)
1
1

sec( x)  1 sec( x)  1
a
Write
c
b as a single fraction.
d
Prove the following identities:
sec(x) cos(x) = 1
csc(x) [1 – cos2(x)] = sin(x)
cot2(x) - csc2(x) = -1
tan(x) csc(x) = sec (x)
sin(x) sec(x) = tan(x)
cos2(x) [sec2(x) - 1] = sin2(x)
cot(x) sec (x) = csc(x)
sec4(x) – tan4(x) = sec2(x) + tan2(x)
Explain why the following are identities. A proof is not necessary. A reasonable explanation will be enough.
sin (–x) = –sin x
cos (–x) = cos x
tan (–x) = –tan x
csc (–x) = –csc x
sec (–x) = sec x
cot (–x) = –cot x
Generate a table of values for the following expressions. Use radians.
x
cos( x)


sin   x 
2

Explain the coincident results.
Generate a table of values for the following expressions. Use radians.
x
tan( x)


cot   x 
2

Explain the coincident results.
The value of a trig function of an angle equals the value of the “cofunction” of the complement of the angle.
Write the 6 cofunction identities:


sin   x  =
2



cos   x  =
2



tan   x  =
2



csc   x  =
2



sec   x  =
2



cot   x  =
2
