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{22} • Trigonometric functions and circular measureExampleFind cos22 1 2◦in surd form.SolutionPutting θ = 22 1 2◦into the double angle formula cos2θ = 2cos 2 θ − 1 and writing x forcos22 1 2◦, we obtaincos45 ◦ = 2x 2 − 112= 2x 2 − 12x 2 = 1 + 22√1 + 2x =2 2 ,which simplifies to 1 2√2 + 2.We can also find a double angle formula for sine using the same diagram.Aθ θ11BDCIn this case, we write down formulas for the area of △ABC in two ways:• On the one hand, the area is given by 1 2 AB · AC sin(∠B AC ) = 1 2 sin2θ.• Since AD = cosθ, we can alternatively split the triangle into two right-angled trianglesand write the area as 2 × 1 2sinθ cosθ = sinθ cosθ.Equating these two expressions for the area, we obtainsin2θ = 2sinθ cosθ.As before, we assumed that θ lies between 0 ◦ and 90 ◦ , but the formula is valid for allvalues of θ.
A guide for teachers – Years 11 and 12 • {23}Exercise 8Find sin15 ◦ in surd form.From the Pythagorean identity and the double angle formula for cosine, we havecos 2 θ + sin 2 θ = 1cos 2 θ − sin 2 θ = cos2θ.Adding these equations and dividing by 2 we obtain cos 2 θ = 1 2(cos2θ + 1), while subtractingthem and dividing by 2 we obtain sin 2 θ = 1 2(1 − cos2θ). These formulas are veryimportant in integral calculus, as discussed in the module The calculus of trigonometricfunctions.Exercise 9Use the double angle formulas for sine and cosine to show thattan2θ =2tanθ1 − tan 2 , for tanθ ≠ ±1.θSummary of double angle formulassin2θ = 2sinθ cosθcos2θ = cos 2 θ − sin 2 θ= 2cos 2 θ − 1= 1 − 2sin 2 θtan2θ =2tanθ1 − tan 2 , for tanθ ≠ ±1θTrigonometric functions of compound anglesIn the previous subsection, we derived formulas for the trigonometric functions of doubleangles. That derivation, which used triangles, was only valid for a limited range ofangles, although the formulas remain true for all angles.In this subsection we find the expansion formulas for sin(A + B), sin(A − B), cos(A + B)and cos(A − B), which are valid for all A and B. The double angle formulas can be recoveredby putting A = B = θ.These formulas are quite central in trigonometry. In the module The calculus of trigonometricfunctions, they are used to find, among other things, the derivative of the sinefunction.
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