Introduction of Trigonometric Functions Calculus:
Trigonometry came from the Greek words ‘trigon-triangle’ and ‘metron-measure’. Trigonometric functions generally define the functions of angle. An equality that satisfies for any value of the variable is generally defined as an identity. An equation is an equality, which is true only for several values of the variable. An identity gives more clear and convenient form than the expression. Trigonometric function calculus are generally applied in modeling periodic phenomena and study of triangles.
Types of Identities in Trigonometric Functions Calculus:
There are several identities in trigonometry function calculus. They are as follows,
Reciprocal identity,
Tangent and Cotangent identity,
Pythagorean identity,
Co-function identity,
Even-Odd identity.
Example Problems for Trigonometric Function Calculus:
Ex 1:
Prove that, sec^2 a + csc^2 a = sec^2 a × csc^2 a, Using Trigonometric Functions calculus.
Proof: L.H.S. = sec^2 a + csc^2 a
=> (1 / cos^2 a) + (1 / sin^2 a) [Using Reciprocal Identity ]
=> (sin^2 a + cos^2 a) / sin^2 a × cos^2 a [Taking L.C.M. ]
=> 1 / sin^2 a × cos^2 a [By Pythagorean Identity]
=> 1 / sin^2 a × 1 / cos^2 a [Separating it Into Two Terms ]
=> csc^2 a × sec^2 a [Using Reciprocal Identity ]
=> sec^2 a × csc^2 a
=> R.H.S.
Hence proved that, sec^2 a + csc^2 a = sec^2 a × csc^2 a
Between, if you have problem on these topics free math tutor online, please browse expert math related websites for more help on 10th cbse sample papers.
Ex 2:
Prove that, sin4 a - 2sin^2 a cos^2 a + cos4 a = cos^2 (2 a), Using Trigonometric Functions calculus.
Proof:
L.H.S. = sin4 a - 2 sin^2 a cos^2 a + cos^4 a
=> (sin^2 a)2 - 2sin^2 a cos^2 a + (cos^2 a)2
=> (sin^2 a - cos^2 a)2 [ Using ( a - b )2 = a2 - 2ab + b2 ]
=> ((1 - cos^2 a) - cos^2 a)2 [By Pythagorean identity ]
=> (1 - 2cos^2 a)2
=>( -(2cos^2 a - 1))2
=>( -cos 2a)2 [ Using Formula: cos 2T = 2 cos^2T - 1 ]
=>cos^2 (2 a)
=> R.H.S.
Hence proved that, sin4 a - 2sin^2 a cos^2 a + cos^4 a = cos^2 (2 a).
Trigonometry came from the Greek words ‘trigon-triangle’ and ‘metron-measure’. Trigonometric functions generally define the functions of angle. An equality that satisfies for any value of the variable is generally defined as an identity. An equation is an equality, which is true only for several values of the variable. An identity gives more clear and convenient form than the expression. Trigonometric function calculus are generally applied in modeling periodic phenomena and study of triangles.
Types of Identities in Trigonometric Functions Calculus:
There are several identities in trigonometry function calculus. They are as follows,
Reciprocal identity,
Tangent and Cotangent identity,
Pythagorean identity,
Co-function identity,
Even-Odd identity.
Example Problems for Trigonometric Function Calculus:
Ex 1:
Prove that, sec^2 a + csc^2 a = sec^2 a × csc^2 a, Using Trigonometric Functions calculus.
Proof: L.H.S. = sec^2 a + csc^2 a
=> (1 / cos^2 a) + (1 / sin^2 a) [Using Reciprocal Identity ]
=> (sin^2 a + cos^2 a) / sin^2 a × cos^2 a [Taking L.C.M. ]
=> 1 / sin^2 a × cos^2 a [By Pythagorean Identity]
=> 1 / sin^2 a × 1 / cos^2 a [Separating it Into Two Terms ]
=> csc^2 a × sec^2 a [Using Reciprocal Identity ]
=> sec^2 a × csc^2 a
=> R.H.S.
Hence proved that, sec^2 a + csc^2 a = sec^2 a × csc^2 a
Between, if you have problem on these topics free math tutor online, please browse expert math related websites for more help on 10th cbse sample papers.
Ex 2:
Prove that, sin4 a - 2sin^2 a cos^2 a + cos4 a = cos^2 (2 a), Using Trigonometric Functions calculus.
Proof:
L.H.S. = sin4 a - 2 sin^2 a cos^2 a + cos^4 a
=> (sin^2 a)2 - 2sin^2 a cos^2 a + (cos^2 a)2
=> (sin^2 a - cos^2 a)2 [ Using ( a - b )2 = a2 - 2ab + b2 ]
=> ((1 - cos^2 a) - cos^2 a)2 [By Pythagorean identity ]
=> (1 - 2cos^2 a)2
=>( -(2cos^2 a - 1))2
=>( -cos 2a)2 [ Using Formula: cos 2T = 2 cos^2T - 1 ]
=>cos^2 (2 a)
=> R.H.S.
Hence proved that, sin4 a - 2sin^2 a cos^2 a + cos^4 a = cos^2 (2 a).
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