Compound-angle formulae

Objective. Expand and apply sin(A±B), cos(A±B) and tan(A±B).

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Learn: the key idea

Compound-angle formulae expand trigonometric functions of a sum or difference of two angles, given by sin(A±B)=sinAcosB±cosAsinB, cos(A±B)=cosAcosB∓sinAsinB, and tan(A±B)=(tanA±tanB)/(1∓tanAtanB). Example: sin75°=sin(45°+30°)=sin45°cos30°+cos45°sin30°=(√6+√2)/4, giving an exact surd value without a calculator. These formulae also prove double-angle identities when A=B.

Worked example

Use a compound-angle formula to find the exact value of cos75°.

1 $Write cos75°=cos(45°+30°)$
2 $Apply cos(A+B)=cosAcosB-sinAsinB$
3 $Substitute exact values: (\sqrt2/2)(\sqrt3/2)-(\sqrt2/2)(1/2)$
4 $Simplify: (\sqrt6-\sqrt2)/4$

Answer: (√6−√2)/4

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Common mistake. Students mix up the sign in cos(A+B), writing cosAcosB+sinAsinB instead of minus; remember cosine's sign flips relative to the given operation, unlike sine's.

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