Important Formulas & Examples: Trigonometric Functions - 1

Important Formulas & Examples: Trigonometric Functions - 1 | Mathematics (Maths) for JEE Main & Advanced PDF Download

``` Page 1

Trigonometric Formula Sheet
De?nition of the Trig Functions
Right Triangle De?nition
Assume that:
0 < ? <
? 2
or 0
< ? < 90
hypotenuse
opposite
? sin? =
opp
hyp
csc? =
hyp
opp
cos? =
hyp
sec? =
hyp
tan? =
opp
cot? =
opp
Unit Circle De?nition
Assume ? can be any angle.
x
y
y
x
1
(x,y)
? sin? =
y
1
csc? =
1
y
cos? =
x
1
sec? =
1
x
tan? =
y
x
cot? =
x
y
Domains of the Trig Functions
sin? , 8 ? 2 (1 ,1 )
cos? , 8 ? 2 (1 ,1 )
tan? , 8 ? 6=
?
n+
1
2
?
? ,wheren2 Z
csc? , 8 ? 6= n? ,wheren2 Z
sec? , 8 ? 6=
?
n+
1
2
?
? ,wheren2 Z
cot? , 8 ? 6= n? ,wheren2 Z
Ranges of the Trig Functions
1? sin? ? 1
1? cos? ? 1
1? tan? ?1
csc?  1 and csc? ? 1
sec?  1 and sec? ? 1
1? cot? ?1
Periods of the Trig Functions
The period of a function is the number, T, such that f (? +T ) = f (? ).
So, if ! is a ?xed number and ? is any angle we have the following periods.
sin(!? ) ) T =
2? !
cos(!? ) ) T =
2? !
tan(!? ) ) T =
? !
csc(!? ) ) T =
2? !
sec(!? ) ) T =
2? !
cot(!? ) ) T =
? !
1
Page 2

Trigonometric Formula Sheet
De?nition of the Trig Functions
Right Triangle De?nition
Assume that:
0 < ? <
? 2
or 0
< ? < 90
hypotenuse
opposite
? sin? =
opp
hyp
csc? =
hyp
opp
cos? =
hyp
sec? =
hyp
tan? =
opp
cot? =
opp
Unit Circle De?nition
Assume ? can be any angle.
x
y
y
x
1
(x,y)
? sin? =
y
1
csc? =
1
y
cos? =
x
1
sec? =
1
x
tan? =
y
x
cot? =
x
y
Domains of the Trig Functions
sin? , 8 ? 2 (1 ,1 )
cos? , 8 ? 2 (1 ,1 )
tan? , 8 ? 6=
?
n+
1
2
?
? ,wheren2 Z
csc? , 8 ? 6= n? ,wheren2 Z
sec? , 8 ? 6=
?
n+
1
2
?
? ,wheren2 Z
cot? , 8 ? 6= n? ,wheren2 Z
Ranges of the Trig Functions
1? sin? ? 1
1? cos? ? 1
1? tan? ?1
csc?  1 and csc? ? 1
sec?  1 and sec? ? 1
1? cot? ?1
Periods of the Trig Functions
The period of a function is the number, T, such that f (? +T ) = f (? ).
So, if ! is a ?xed number and ? is any angle we have the following periods.
sin(!? ) ) T =
2? !
cos(!? ) ) T =
2? !
tan(!? ) ) T =
? !
csc(!? ) ) T =
2? !
sec(!? ) ) T =
2? !
cot(!? ) ) T =
? !
1
Page 3

Trigonometric Formula Sheet
De?nition of the Trig Functions
Right Triangle De?nition
Assume that:
0 < ? <
? 2
or 0
< ? < 90
hypotenuse
opposite
? sin? =
opp
hyp
csc? =
hyp
opp
cos? =
hyp
sec? =
hyp
tan? =
opp
cot? =
opp
Unit Circle De?nition
Assume ? can be any angle.
x
y
y
x
1
(x,y)
? sin? =
y
1
csc? =
1
y
cos? =
x
1
sec? =
1
x
tan? =
y
x
cot? =
x
y
Domains of the Trig Functions
sin? , 8 ? 2 (1 ,1 )
cos? , 8 ? 2 (1 ,1 )
tan? , 8 ? 6=
?
n+
1
2
?
? ,wheren2 Z
csc? , 8 ? 6= n? ,wheren2 Z
sec? , 8 ? 6=
?
n+
1
2
?
? ,wheren2 Z
cot? , 8 ? 6= n? ,wheren2 Z
Ranges of the Trig Functions
1? sin? ? 1
1? cos? ? 1
1? tan? ?1
csc?  1 and csc? ? 1
sec?  1 and sec? ? 1
1? cot? ?1
Periods of the Trig Functions
The period of a function is the number, T, such that f (? +T ) = f (? ).
So, if ! is a ?xed number and ? is any angle we have the following periods.
sin(!? ) ) T =
2? !
cos(!? ) ) T =
2? !
tan(!? ) ) T =
? !
csc(!? ) ) T =
2? !
sec(!? ) ) T =
2? !
cot(!? ) ) T =
? !
1
Identities and Formulas
Tangent and Cotangent Identities
tan? =
sin? cos? cot? =
cos? sin? Reciprocal Identities
sin? =
1
csc? csc? =
1
sin? cos? =
1
sec? sec? =
1
cos? tan? =
1
cot? cot? =
1
tan? Pythagorean Identities
sin
2
? +cos
2
? =1
tan
2
? +1=sec
2
? 1+cot
2
? =csc
2
? Even and Odd Formulas
sin( ? )= sin? cos( ? )=cos? tan( ? )= tan? csc( ? )= csc? sec( ? )=sec? cot( ? )= cot? Periodic Formulas
If n is an integer
sin(? +2? n) = sin? cos(? +2? n)=cos? tan(? +? n)=tan? csc(? +2? n)=csc? sec(? +2? n)=sec? cot(? +? n)=cot? Double Angle Formulas
sin(2? )=2sin? cos? cos(2? )=cos
2
?  sin
2
? =2cos
2
?  1
=1 2sin
2
? tan(2? )=
2tan? 1 tan
2
If x is an angle in degrees and t is an angle in
? 180
=
t
x
) t =
? x
180
and x =
180
t
? Half Angle Formulas
sin? =±
r
1 cos(2? )
2
cos? =±
r
1+cos(2? )
2
tan? =±
s
1 cos(2? )
1+cos(2? )
Sum and Di? erence Formulas
sin(? ± )=sin? cos ±cos? sin cos(? ± )=cos? cos ? sin? sin tan(? ± )=
tan? ±tan 1? tan? tan Product to Sum Formulas
sin? sin =
1
2
[cos(?   ) cos(? + )]
cos? cos =
1
2
[cos(?   )+cos(? + )]
sin? cos =
1
2
[sin(? + )+sin(?   )]
cos? sin =
1
2
[sin(? + ) sin(?   )]
Sum to Product Formulas
sin? +sin =2sin
?
? + 2
?
cos
?
?   2
?
sin?  sin =2cos
?
? + 2
?
sin
?
?   2
?
cos? +cos =2cos
?
? + 2
?
cos
?
?   2
?
cos?  cos = 2sin
?
? + 2
?
sin
?
?   2
?
Cofunction Formulas
sin
?
? 2
? ?
=cos? csc
?
? 2
? ?
=sec? tan
?
? 2
? ?
=cot? cos
?
? 2
? ?
=sin? sec
?
? 2
? ?
=csc? cot
?
? 2
? ?
=tan? 2
Page 4

Trigonometric Formula Sheet
De?nition of the Trig Functions
Right Triangle De?nition
Assume that:
0 < ? <
? 2
or 0
< ? < 90
hypotenuse
opposite
? sin? =
opp
hyp
csc? =
hyp
opp
cos? =
hyp
sec? =
hyp
tan? =
opp
cot? =
opp
Unit Circle De?nition
Assume ? can be any angle.
x
y
y
x
1
(x,y)
? sin? =
y
1
csc? =
1
y
cos? =
x
1
sec? =
1
x
tan? =
y
x
cot? =
x
y
Domains of the Trig Functions
sin? , 8 ? 2 (1 ,1 )
cos? , 8 ? 2 (1 ,1 )
tan? , 8 ? 6=
?
n+
1
2
?
? ,wheren2 Z
csc? , 8 ? 6= n? ,wheren2 Z
sec? , 8 ? 6=
?
n+
1
2
?
? ,wheren2 Z
cot? , 8 ? 6= n? ,wheren2 Z
Ranges of the Trig Functions
1? sin? ? 1
1? cos? ? 1
1? tan? ?1
csc?  1 and csc? ? 1
sec?  1 and sec? ? 1
1? cot? ?1
Periods of the Trig Functions
The period of a function is the number, T, such that f (? +T ) = f (? ).
So, if ! is a ?xed number and ? is any angle we have the following periods.
sin(!? ) ) T =
2? !
cos(!? ) ) T =
2? !
tan(!? ) ) T =
? !
csc(!? ) ) T =
2? !
sec(!? ) ) T =
2? !
cot(!? ) ) T =
? !
1
Identities and Formulas
Tangent and Cotangent Identities
tan? =
sin? cos? cot? =
cos? sin? Reciprocal Identities
sin? =
1
csc? csc? =
1
sin? cos? =
1
sec? sec? =
1
cos? tan? =
1
cot? cot? =
1
tan? Pythagorean Identities
sin
2
? +cos
2
? =1
tan
2
? +1=sec
2
? 1+cot
2
? =csc
2
? Even and Odd Formulas
sin( ? )= sin? cos( ? )=cos? tan( ? )= tan? csc( ? )= csc? sec( ? )=sec? cot( ? )= cot? Periodic Formulas
If n is an integer
sin(? +2? n) = sin? cos(? +2? n)=cos? tan(? +? n)=tan? csc(? +2? n)=csc? sec(? +2? n)=sec? cot(? +? n)=cot? Double Angle Formulas
sin(2? )=2sin? cos? cos(2? )=cos
2
?  sin
2
? =2cos
2
?  1
=1 2sin
2
? tan(2? )=
2tan? 1 tan
2
If x is an angle in degrees and t is an angle in
? 180
=
t
x
) t =
? x
180
and x =
180
t
? Half Angle Formulas
sin? =±
r
1 cos(2? )
2
cos? =±
r
1+cos(2? )
2
tan? =±
s
1 cos(2? )
1+cos(2? )
Sum and Di? erence Formulas
sin(? ± )=sin? cos ±cos? sin cos(? ± )=cos? cos ? sin? sin tan(? ± )=
tan? ±tan 1? tan? tan Product to Sum Formulas
sin? sin =
1
2
[cos(?   ) cos(? + )]
cos? cos =
1
2
[cos(?   )+cos(? + )]
sin? cos =
1
2
[sin(? + )+sin(?   )]
cos? sin =
1
2
[sin(? + ) sin(?   )]
Sum to Product Formulas
sin? +sin =2sin
?
? + 2
?
cos
?
?   2
?
sin?  sin =2cos
?
? + 2
?
sin
?
?   2
?
cos? +cos =2cos
?
? + 2
?
cos
?
?   2
?
cos?  cos = 2sin
?
? + 2
?
sin
?
?   2
?
Cofunction Formulas
sin
?
? 2
? ?
=cos? csc
?
? 2
? ?
=sec? tan
?
? 2
? ?
=cot? cos
?
? 2
? ?
=sin? sec
?
? 2
? ?
=csc? cot
?
? 2
? ?
=tan? 2
Unit Circle
0
,2? (1,0)
180
,? ( 1,0)
(0,1)
90
,
? 2
(0, 1)
270
,
3? 2
30
,
? 6
(
p 3
2
,
1
2
)
45
,
? 4
(
p 2
2
,
p 2
2
)
60
,
? 3
(
1
2
,
p 3
2
)
120
,
2? 3
( 1
2
,
p 3
2
)
135
,
3? 4
( p 2
2
,
p 2
2
)
150
,
5? 6
( p 3
2
,
1
2
)
210
,
7? 6
( p 3
2
, 1
2
)
225
,
5? 4
( p 2
2
, p 2
2
)
240
,
4? 3
( 1
2
, p 3
2
)
300
,
5? 3
(
1
2
, p 3
2
)
315
,
7? 4
(
p 2
2
, p 2
2
)
330
,
11? 6
(
p 3
2
, 1
2
)
For any ordered pair on the unit circle (x,y):cos? = x and sin? = y
Example
cos(
7? 6
)= p 3
2
sin(
7? 6
)= 1
2
3
Page 5

Trigonometric Formula Sheet
De?nition of the Trig Functions
Right Triangle De?nition
Assume that:
0 < ? <
? 2
or 0
< ? < 90
hypotenuse
opposite
? sin? =
opp
hyp
csc? =
hyp
opp
cos? =
hyp
sec? =
hyp
tan? =
opp
cot? =
opp
Unit Circle De?nition
Assume ? can be any angle.
x
y
y
x
1
(x,y)
? sin? =
y
1
csc? =
1
y
cos? =
x
1
sec? =
1
x
tan? =
y
x
cot? =
x
y
Domains of the Trig Functions
sin? , 8 ? 2 (1 ,1 )
cos? , 8 ? 2 (1 ,1 )
tan? , 8 ? 6=
?
n+
1
2
?
? ,wheren2 Z
csc? , 8 ? 6= n? ,wheren2 Z
sec? , 8 ? 6=
?
n+
1
2
?
? ,wheren2 Z
cot? , 8 ? 6= n? ,wheren2 Z
Ranges of the Trig Functions
1? sin? ? 1
1? cos? ? 1
1? tan? ?1
csc?  1 and csc? ? 1
sec?  1 and sec? ? 1
1? cot? ?1
Periods of the Trig Functions
The period of a function is the number, T, such that f (? +T ) = f (? ).
So, if ! is a ?xed number and ? is any angle we have the following periods.
sin(!? ) ) T =
2? !
cos(!? ) ) T =
2? !
tan(!? ) ) T =
? !
csc(!? ) ) T =
2? !
sec(!? ) ) T =
2? !
cot(!? ) ) T =
? !
1
Identities and Formulas
Tangent and Cotangent Identities
tan? =
sin? cos? cot? =
cos? sin? Reciprocal Identities
sin? =
1
csc? csc? =
1
sin? cos? =
1
sec? sec? =
1
cos? tan? =
1
cot? cot? =
1
tan? Pythagorean Identities
sin
2
? +cos
2
? =1
tan
2
? +1=sec
2
? 1+cot
2
? =csc
2
? Even and Odd Formulas
sin( ? )= sin? cos( ? )=cos? tan( ? )= tan? csc( ? )= csc? sec( ? )=sec? cot( ? )= cot? Periodic Formulas
If n is an integer
sin(? +2? n) = sin? cos(? +2? n)=cos? tan(? +? n)=tan? csc(? +2? n)=csc? sec(? +2? n)=sec? cot(? +? n)=cot? Double Angle Formulas
sin(2? )=2sin? cos? cos(2? )=cos
2
?  sin
2
? =2cos
2
?  1
=1 2sin
2
? tan(2? )=
2tan? 1 tan
2
If x is an angle in degrees and t is an angle in
? 180
=
t
x
) t =
? x
180
and x =
180
t
? Half Angle Formulas
sin? =±
r
1 cos(2? )
2
cos? =±
r
1+cos(2? )
2
tan? =±
s
1 cos(2? )
1+cos(2? )
Sum and Di? erence Formulas
sin(? ± )=sin? cos ±cos? sin cos(? ± )=cos? cos ? sin? sin tan(? ± )=
tan? ±tan 1? tan? tan Product to Sum Formulas
sin? sin =
1
2
[cos(?   ) cos(? + )]
cos? cos =
1
2
[cos(?   )+cos(? + )]
sin? cos =
1
2
[sin(? + )+sin(?   )]
cos? sin =
1
2
[sin(? + ) sin(?   )]
Sum to Product Formulas
sin? +sin =2sin
?
? + 2
?
cos
?
?   2
?
sin?  sin =2cos
?
? + 2
?
sin
?
?   2
?
cos? +cos =2cos
?
? + 2
?
cos
?
?   2
?
cos?  cos = 2sin
?
? + 2
?
sin
?
?   2
?
Cofunction Formulas
sin
?
? 2
? ?
=cos? csc
?
? 2
? ?
=sec? tan
?
? 2
? ?
=cot? cos
?
? 2
? ?
=sin? sec
?
? 2
? ?
=csc? cot
?
? 2
? ?
=tan? 2
Unit Circle
0
,2? (1,0)
180
,? ( 1,0)
(0,1)
90
,
? 2
(0, 1)
270
,
3? 2
30
,
? 6
(
p 3
2
,
1
2
)
45
,
? 4
(
p 2
2
,
p 2
2
)
60
,
? 3
(
1
2
,
p 3
2
)
120
,
2? 3
( 1
2
,
p 3
2
)
135
,
3? 4
( p 2
2
,
p 2
2
)
150
,
5? 6
( p 3
2
,
1
2
)
210
,
7? 6
( p 3
2
, 1
2
)
225
,
5? 4
( p 2
2
, p 2
2
)
240
,
4? 3
( 1
2
, p 3
2
)
300
,
5? 3
(
1
2
, p 3
2
)
315
,
7? 4
(
p 2
2
, p 2
2
)
330
,
11? 6
(
p 3
2
, 1
2
)
For any ordered pair on the unit circle (x,y):cos? = x and sin? = y
Example
cos(
7? 6
)= p 3
2
sin(
7? 6
)= 1
2
3
```

Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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