About Double Angle Formulas
The trigonometric ratios of double angles (2) are expressed in terms of trigonometric ratios of single angles () using double angle formulae. The Pythagorean identities are used to derive some alternative formulas. The double angle formulas are particular cases of (and so derived from) the sum formulas of trigonometry. Let's review the trigonometry sum formulas.
- sin(A + B) = sin A cos B + cos A sin B
- cos(A + B) = cos A cos B - Sin A sin B
- (A + B) = (tan A + tan B)/(1 - tan A tan B)
Define Double Angle Formulas
On substituting A = B in each of the preceding sum formulas, we may get the double angle formulas for sin, cos, and tan. We will also use the Pythagorean identities to derive some alternative formulas. The double angle formulas are listed here, along with their derivations.
Double Angle Formulas
Double angle formulas of sin, cos, and tan are:
- sin 2A = 2 sin A cos A (or)(2 tan A)/(1 + tan2)
- cos 2A = cos2A - sin2A (or) 2cos2 A - 1(or) 1 - 2sin2A (or)(1 -tan<2A) / (1 + tan2A)
- tan2A = (2 tan A)/(1 - tan2)
Derivation of double Angle Formulas
Now derive double angle formula(s) of cos:
Double Angle Formulas of Cos
The sum formula of cosine function is,
cos (A + B) = cos A cos B - sin A Sin B
When A = B, the above formula becomes,
cos (A + A) = cos A cos A - sin A Sin A
cos 2A = cos A2 - sin2A
Let us take this as a starting point to generate two more cos 2A formulas using the Pythagorean identity
sin2A + cos2A = 1
- cos 2A = cos2A − (1 − cos2A) =2cos2A - 1
- cos 2A = (1- sin2A) - sin2A = 1 - 2sin2A
Using the basic formula, we can now obtain the expression for cos 2A in terms of tan.
cos2A=cos2A − sin2A = cos2A(1−sin2A/cos2A)=1/sec2A(1−tan2A)=1/1+tan2A(1−tan2A)= (1 - tan2A) /(1 +tan2A)
Hence, Double angle formulasof cosine function are:
cos 2A = cos2A - sin2A (or) 2cos2A - 1 (or) 1 - 2sin2A (or) (1 - tan2A) /(1 +tan2A)
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