14-7 Double-Angle and Half-Angle Identities

Objectives

To verify and use double-angle identities

To verify and use half-angle identities

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If an equation contains two variables, such as radius and height, you can find a special case of the equation by replacing one of the variables with the other variable.

Essential Understanding The double-angle identities are special cases of the angle sum identities. Substitute θ 2 theta over 2 for θ theta in certain double-angle identities and you get the half-angle identities.

Here's Why It Works

Let θ = A = B . theta equals eh equals b .

cos ( A + B ) = cos A cos B − sin A sin B Cosine Angle Sum Identity cos ( θ + θ ) = cos θ cos θ − sin θ sin θ Substitute  θ  for  A  and  B . cos 2 θ = cos 2 θ − sin 2 θ Simplify . table with 3 rows and 3 columns , row1 column 1 , cosine . open , eh plus b , close , column 2 equals cosine eh cosine b minus sine eh sine b , column 3 cap cosinecap anglecap sumcap identity , row2 column 1 , cosine . open , theta plus theta , close , column 2 equals cosine theta cosine theta minus sine theta sine theta , column 3 cap substitute . theta , for eh , and b . , row3 column 1 , cosine 2 theta , column 2 equals , cosine squared , theta minus , sine squared , theta , column 3 cap simplify , . , end table

You can make the same substitution in the other angle sum identities.

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