Solving Trigonometric Identities Online
An equation is an equality which is true only for several values of the variable
. An expression can be given more clear and convenient form by an identity.
Solving Trigonometric Identities Online:
Online trigonometric identities solving shows a clear idea about a tutor solves a problem in online. Some of the trigonometric identities and example problems using trigonometric identities are listed below.
Types of Trigonometric Identities:
There are several identities in trigonometry. Here we see some of trignometric identities online. They are as follows,
Reciprocal identities,
sin x = '1 / csc x' , csc x = '1 / sin x' ,
cos x = '1 / sec x' , sec x = '1 / cos x' ,
tan x = '1 / cot x' , cot x = '1 / tan x' .
Tangent and Cotangent identities,
tan x = 'sin x / cos x' ,
cot x = 'cos x / sin x' .
Pythagorean identities,
sin2x + cos2x = 1,
1 + tan2x = sec2x,
1 + cot2x = csc2x.
Co-function identities,
sin (90 x) = cos x, cos (90 x) = sin x, tan (90 x) = cot x,
csc (90 x) = sec x, sec (90 x) = csc x, cot (90 x) = tan x.
Even-Odd identities,
sin (x) = sin x, cos (x) = cos x, tan (x) = tan x,
csc (x) = csc x, sec (x) = sec x, cot (x) = cot x.
Examples of Solving Trigonometric Identities Online:
Ex 1: Solve sin30 by using cos60 = 0.5
Sol : Solving sin 30,
sin30 = sin(90 60)
= cos60 [Using Co-Function Identity]
= 0.5
Therefore, sin30 = 0.5
Ex 2: Prove that, sec2x + csc2x = sec2x csc2x using trigonometric identities.
Proof:
Solving L.H.S. ,
L.H.S. = sec2x + csc2x
= ('1 / (cos^(2)x)' ) + ('1 / (sin^(2)x)' ) [Using Reciprocal Identity]
= ' (sin^(2)x + cos^(2)x) / (sin^(2)x cos^(2)x)' [Taking L.C.M. ]
= '1 / (sin^(2)x cos^(2)x)' [Using Pythagorean Identity]
= '1 / (sin^(2)x)' '1 / (cos^(2)x)' [Separating it Into Two Terms]
= csc2x sec2x [Using Reciprocal Identity]
= sec2x csc2x
= R.H.S.
Hence, proved that sec2x + csc2x = sec2x csc2x.
Ex 3:
Prove that, sin4 a 2sin2 a cos2 a + cos4 a = cos2 (2 a), Using Trigonometric Functions calculus.
Proof:
L.H.S. = sin4 a 2 sin2 a cos2 a + cos4 a
= (sin2 a)2 2sin2 a cos2 a + (cos2 a)2
= (sin2 a cos2 a)2 [ Using ( a b )2 = a2 2ab + b2 ]
= ((1 cos2 a) cos2 a)2 [By Pythagorean identity ]
= (1 2cos2 a)2
=( (2cos2 a 1))2
=( cos 2a)2 [ Using Formula: cos 2 = 2 cos2 1 ]
=cos2 (2 a)
= R.H.S.
Hence proved that, sin4 a 2sin2 a cos2 a + cos4 a = cos2 (2 a).
by: Smith
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