cotangent formulas
TRIGONOMETRY
NAME____________________
Date__________Per.________
Deriving a formula for cot(a+b).
1. First, identify each sum formula:
2.
a) Write what cot
means in terms of
sin and cos.
3.
a) Substitute your
formulas from #1 in
the numerator and
denominator of the
quotient in #2b).
= cos(α)cos(β) – sin(α)sin(β)
= sin(α)cos(β) + cos(α)sin(β)
b) Express
cot(α + β)
in terms of sin
and cos.
cot(θ) =
cot(α + β)
cot(α + β) =
–
__________________________
=
+
cos(α)cos(β)
b) Divide every
term in the top and
the bottom by:
sin(α)sin(β)
–
= __________________________
sin(α)sin(β)
sin(α)cos(β)
cos(α)sin(β)
+
c) Cancel terms that
reduce to “1”.
–
= ____________________
+
d) Rewrite each cos/sin (of the same angle
only) as cot. This is the desired formula.
cot(α + β) =
4.
Fill in () to prove that:
cot(α – β) = cot(α + -β) =
cot( )cot( ) – 1
cot( ) + cot( )
5.
Find the double angle
formula for cotangent:
- cot( )cot( ) – 1
=
=
- cot( ) + cot( )
cot(2θ) = cot( +
)=
cot( )cot( )+1
cot( ) – cot( )
cot( )cot( ) – 1
cot( ) + cot( )
=