As you know there are these trigonometric formulas like Sin 2x, Cos 2x, Tan 2x which are known as double angle formulae for they have double angles in them. To get a good understanding of this topic, Let’s go through the practice examples provided.
Cos 2 X = Cos2A – Sin2A
= 2Cos2A – 1
= 1 – 2Cos2A
Introduction to Cos 2 Theta formula
Let’s have a look at trigonometric formulae known as the double angle formulae. They are said to be so as it involves double angles trigonometric functions, i.e. Cos 2x.
Deriving Double Angle Formulae for Cos 2t
Let’s start by considering the addition formula.
Cos(A + B) = Cos A cos B – Sin A sin B
Let’s equate B to A, i.e A = B
And then, the first of these formulae becomes: Cos(t + t) = Cos t cos t – Sin t sin t
so that Cos 2t = Cos2t – Sin2t
And this is how we get second double-angle formula, which is so called because you are doubling the angle (as in 2A).
Practice Example for Cos 2:
Solve the equation cos 2a = sin a, for – Π \(\leq\) a< Π
Solution: Let’s use the double angle formula cos 2a = 1 − 2 sin2 a
It becomes 1 − 2 sin2 a = sin a
2 sin2 a + sin a − 1=0,
Let’s factorise this quadratic equation with variable sinx
(2 sin a − 1)(sin a + 1) = 0
2 sin a − 1 = 0 or sin a + 1 = 0
sin a = 1/2 or sin a = −1
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