Formulae For Trigonometric Functions & Inverse ...
[Pages:5]Formulae For Trigonometric Functions & Inverse Trigonometric Functions
Trigonometric Formulae:
Relation between trigonometric ratios
sin a) tan
cos
1 b) tan
cot
cos d) cot
sin
1 e) cosec
sin
c) tan .cot 1
1 f) sec
cos
Trigonometric identities
a) sin2 cos2 1 b) 1 tan2 sec2 c) 1 cot2 cosec2
Addition / subtraction formulae & some related results
Multiple angle formulae involving 2A and 3A
a) sin 2A 2sin Acos A
AA b) sin A 2sin cos
22 c) cos2 A cos2 A sin2 A
a) sin A B sin Acos B cos Asin B b) cos A B cos Acos B sin Asin B c) cos A B cos A B cos2 A sin2 B cos2 B sin2 A d) sin A Bsin A B sin2 A sin2 B cos2 B cos2 A
d) cos A cos2 A sin2 A
2
2
e) cos 2 A 2cos2 A 1
f) 2cos2 A 1 cos2 A
e) tan A B tan A tan B
1 tan A tan B
f) cot A B cot B cot A 1
cot B cot A Transformation of sums / differences into products & vice-versa
g) cos 2 A 1 2sin2 A h) 2sin2 A 1 cos 2A
2 tan A i) sin 2 A
1 tan2 A
a) sinC sin D 2sin C D cos C D
2
2
b) sinC sin D 2cos C D sin C D
2
2
c) cosC cos D 2 cos C D cos C D
2
2
d) cosC cos D 2sin C D sin C D
2
2
e) 2sin Acos B sin A B sin A B
f) 2cos Asin B sin A B sin A B
g) 2cos AcosB cos A B cos A B
h) 2sin Asin B cos A B cos A B
1 tan2 A j) cos 2 A
1 tan2 A 2 tan A
k) tan 2 A 1 tan2 A
l) sin 3A 3sin A 4sin3 A
m) cos 3A 4cos3 A 3cos A 3tan A tan3 A
n) tan 3A 1 3tan2 A
Relations in Different Measures of Angle
Angle in Radian Measure = Angle in Degree Measure?
180 180
Angle in Degree Measure = Angle in Radian Measure?
l (in radian measure) r
Also followings are of importance as well:
1Right angle 90o
1 o = 60, 1 = 60
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List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering)
1o = = 0.01745 radians approximately
180
1 radian = 57o1745 or 206265 seconds .
General Solutions
a) sin x sin y x n (1 )n y, where n Z . b) cos x cos y x 2n y, where n Z . c) tan x tan y x n y, where n Z .
Relation in Degree & Radian Measures
Angles in Degree 0 Angles in Radian 0c
30
c
6
45
c
4
60
c
3
90
c
2
180 270 360
c
c
3
2
2 c
In actual practice, we omit the exponent `c' and instead of writing c we simply write and similarly for others.
Trigonometric Ratio of Standard Angles
Degree /Radian
0
30
45
60
90
T ? Ratios
0
6
4
3
2
sin
0
1
1
3
1
2
2
2
cos
1
3
1
1
0
2
2
2
tan
0
1
1
3
3
cosec
2
2 2
1
3
sec
1
2
2
2
3
cot
3
1
1
0
3
Trigonometric Ratios of Allied Angles
Angles
T- Ratios
2
2
sin
cos cos sin
sin
3
2
cos
3
2
cos
2 OR
sin
2
sin
cos
sin sin cos cos sin sin cos cos
tan
cot cot tan tan cot cot tan tan
cot
tan tan cot cot tan tan cot cot
sec
cosec cosec sec sec cosec cosec sec sec
cosec
sec sec cosec cosec sec sec cosec cosec
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MATHEMATICS ? List Of Formulae for Class XII By OP Gupta (+91-9650350480)
Inverse Trigonometric Formulae:
01.
a)
sin1
x
cosec1
1
,
x
1,1
x
c)
cos1 x
sec1
1 x
,
x
1,1
b)
cosec1 x
sin1
1
,
x
, 1 1,
x
d) sec1
x
cos1
1 x
,
x
, 1 1,
e)
tan1
x
cot1
1 x
,
x
0
cot1
1 x
,
x
0
f)
cot
1
x
tan1
1 x
,
x
0
tan
1
1 x
,
x
0
02. a) sin1 x sin1 x, x 1,1 c) tan1 x tan1 x, x R e) sec1 x sec1 x, | x | 1
b) cos1 x cos1 x, x 1,1 d) cosec1 x cosec1x, | x | 1 f) cot1 x cot1 x, x R
03.
a) sin1 sin x
x ,
x
2
2
c) tan1 tan x x, x
22
e) sec1 sec x x, 0 x , x
2
b) cos1 cos x x, 0 x
d) cosec1 cosec x x, x , x 0
2
2
f) cot1 cot x x, 0 x
04.
a) sin1 x cos1 x
,
x 1,1
2
b) tan1 x cot1 x
,
x R
2
c) cosec1x sec1 x , | x | 1 i.e., x 1 or x 1 2
05. a) sin1 x sin1 y sin1 x 1 y2 y 1 x2
b) cos1 x cos1 y cos1 xy 1 x2 1 y2
tan1
x y 1 xy
,
xy 1
c) tan1 x tan1 y
tan
1
x y 1 xy
,
x
0,
y 0,
xy 1
tan1
x y 1 xy
,
x 0,
y
0,
xy
1
tan 1
xy 1 xy
,
xy 1
d)
tan
1
x
tan
1
y
tan
1
x y 1 xy
,
x 0,
y 0,
xy 1
tan1
x
y
,
x 0,
y 0,
xy 1
1 xy
e)
tan1
x
tan1
y
tan1
z
tan 1
x y 1 xy
z xyz yz zx
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List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering)
06.
a)
2
tan1
x
sin1
1
2x x2
,
|
x
|
1
b) 2 tan1
x
cos1
1 1
x2 x 2
,
x
0
c)
2
tan 1
x
tan1
2x 1 x2
,
1
x
1
07. Principal Value: Numerically smallest angle is known as the principal value.
Finding the principal value: For finding the principal value, following algorithm can be followed?
STEP1? Firstly, draw a trigonometric circle and mark the quadrant in which the angle may lie. STEP2? Select anticlockwise direction for 1st and 2nd quadrants and clockwise direction for 3rd and
4th quadrants. STEP3? Find the angles in the first rotation. STEP4? Select the numerically least (magnitude wise) angle among these two values. The angle
thus found will be the principal value. STEP5? In case, two angles one with positive sign and the other with the negative sign qualify for
the numerically least angle then, it is the convention to select the angle with positive sign as principal value.
The principal value is never numerically greater than .
08. Table demonstrating domains and ranges of Inverse Trigonometric functions:
Inverse Trigonometric Functions i.e., f ( x) Domain/ Values of x Range/ Values of f ( x)
sin1 x
[1, 1]
2
,
2
cos1 x
[1, 1]
[0, ]
cosec1 x sec1 x tan 1 x cot 1x
R (1, 1) R (1, 1)
R R
2
,
2
{0}
[0, ]
2
2
,
2
(0, )
Discussion about the range of inverse circular functions
other than their respective principal value branch
We know that the domain of sine function is the set of real numbers and
range
is
the
closed
interval
[?1,
1].
If
we
restrict
its
domain
to
3 ,
2
2
,
2
,
2
,
3 2 , 2
etc.
then,
it
becomes
bijective
with
the
range
[?1,
1].
So, we can define the inverse of sine function in each of these intervals. Hence, all the intervals of sin?1 function, except principal value branch
(here except
of
2
,
2
for
sin?1
function)
are known as the
range
of
sin?1
other than its principal value branch. The same discussion can be
extended for other inverse circular functions.
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MATHEMATICS ? List Of Formulae for Class XII By OP Gupta (+91-9650350480)
09. To simplify inverse trigonometrical expressions, following substitutions can be considered:
Expression
a2 x2 or a2 x2
Substitution x a tan or x a cot
a2 x2 or a2 x2
x a sin or x a cos
x2 a2 or x2 a2
x a sec or x a cosec
ax
ax
or
ax
a x
a2 x2
a2 x2
or
a2 x2
a2 x2
x
ax
or
a x
x
x
ax
or
a x
x
x a cos 2 x2 a2 cos 2 x a sin2 or x a cos2 x a tan2 or x a cot2
Note the followings and keep them in mind: The symbol sin1x is used to denote the smallest angle whether positive or negative, the sine of this angle will give us x. Similarly cos1x, tan1x, cosec1x, sec1x, and cot1x are defined.
You should note that sin1x can be written as arcsinx . Similarly other Inverse Trigonometric Functions can also be written as arccosx, arctanx, arcsecx etc.
Also note that sin1x (and similarly other Inverse Trigonometric Functions) is entirely different from (sin x)1 . In fact, sin1x is the measure of an angle in Radians whose sine is x whereas (sin x)1 is 1 (which is obvious as per the laws of exponents).
sin x
Keep in mind that these inverse trigonometric relations are true only in their domains i.e., they are valid only for some values of `x' for which inverse trigonometric functions are well defined!
Hii, All! I hope this texture may have proved beneficial for you. While going through this material, if you noticed any error(s) or, something which doesn't make sense to you, please bring it in my notice through SMS or Call at +91-9650 350 480 or Email at theopgupta@.
With lots of Love & Blessings!
- OP Gupta [+91-9650 350 480] Electronics & Communications Engineering, Indira Award Winner theOPGupta.
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