Trigonometric functions and circular measure - the Australian ...

A guide for teachers – Years 11 and 12 • {21}Double angle formulasThe sine and cosine functions are not linear. For example, cos(A + B) ≠ cos A + cosB.The correct formula for cos(A + B) is given in the next subsection. In the special casewhen A = B = θ, we would like to obtain a simple formula for cos(2θ) = cos2θ, called thedouble angle formula. It is particularly useful in applications and in calculus and is quiteeasy to derive.Consider the following isosceles triangle ABC , with sides AB and AC both of length 1and apex angle 2θ. We will need to assume, for the present, that 0 ◦ < θ < 90 ◦ .Aθ θ11BDCLet AD be the perpendicular bisector of the interval BC . Then, using basic properties ofan isosceles triangle, we know that AD bisects ∠B AC and so BD = DC = sinθ. Applyingthe cosine rule to the triangle ABC , we immediately havecos2θ = 12 + 1 2 − 4sin 2 θ2 × 1 × 1= 2 − 4sin2 θ2= 1 − 2sin 2 θ.Replacing 1 by cos 2 θ + sin 2 θ, we arrive at the double angle formulacos2θ = cos 2 θ − sin 2 θ.This may also be written as cos2θ = 2cos 2 θ − 1 or cos2θ = 1 − 2sin 2 θ, but is best learntin the form given above.We derived this formula under the assumption that θ was between 0 ◦ and 90 ◦ , but theformula in fact holds for all values of θ. We shall see this in the next subsection, wherewe prove the general expansion formula for cos(A + B).