# Trigonometric Ratios of Compound Angles ## Trigonometric Ratios of Compound Angles

The sum or difference of two or more angles is said to be a compound angle. If A and B are two angles then A + B or A - B are known as the compound angles.

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We express the trigonometrical ratios of compound angle A + B and A - B in terms of trigonometrical ratios of the angles A and B.

The trigonometrical ratios of the angles A + B and A - B are known as the addition and subtraction formula respectively.

### Trigonometric Ratios of Compound Angle A + B (Addition Formula)

Let a revolving line start from OX and trace out an angle XOY = A and revolve further through an angle YOZ = B. Therefore, XOZ = A + B. Let, P be any point on OZ. Draw PM perpendicular to OX and PN perpendicular to OY. From N draw NQ perpendicular to OX and NR perpendicular to PM.

Here, RPN = 90° - PNR     [  NRMP]

= RNO   [  PNOY]

= NOQ   [  RN OX]

= A

Again, RMQN is a rectangle. So, MR = QN and RN = MQ.

Now from right angle ΔOMP, #### Tangent Formula of Compound Angle (A + B) #### Cotangent Formula of Compound Angle (A + B) ### Trigonometric Ratios of Compound Angle A – B (Subtraction Formula)

Let a revolving line start from OX and trace out an angle XOY = A and then revolve back through an angle YOZ = B. Therefore, XOZ = A - B. Let, P be any point on the line OZ. Draw PM perpendicular to OX and PN perpendicular to OY. From N draw NQ perpendicular to OX and NR perpendicular to MP produced.

Here, RPN = 90° - PNR     [  PRNR]

= RNY    [  PNOY]

= XOY   [  OX NR]

= A

Again, QMRN is a rectangle. So, QN = MR and QM = NR.

Now from right angle ΔOMP, #### Tangent Formula of Compound Angle (A – B) #### Cotangent Formula of Compound Angle (A - B) ### List of Trigonometric Formula for Compound Angles A + B and A – B ### Worked Out Examples       #### Do you have any questions regarding the trigonometric ratios of compound angles?

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