Multiple and Sub Multiple Angles

Multiple and Sub Multiple Angles

Angle is the pivot around which the topic of Trigonometry revolves. Trigonometry studies angles and their relationship. When there is a single function or a single angle, the computation is comparatively easy. But there are various formulae for multiples and sub multiples of angles too. These multiple and sub multiple angles formula should rather be called as identities as they hold true for all angles. These formulae prove useful in solving intricate trigonometric equations.

It is also possible to find the trigonometric ratios of negative angles, multiple and sub multiple of an angle or compound angles. The coming sections illustrate trigonometric ratios of multiple and sub multiple angles along with various examples. Drafted below are the various topics covered under this head in the following sections:

  • Trigonometric Ratios of Compound Angles
  • Trigonometric Ratios of Submultiple of an angle
  • Trigonometric Ratios for Negative Angles
  • Solved examples on Trigonometric Ratio

We shall just give an outline of these topics here. Those interested in going into the intricacies of the topics can refer the following sections.

Compound angles:

Angles composed of an algebraic sum or difference of two or more angles are called compound angles. Trigonometric ratios of compound angles include evaluation of trigonometric sum or trigonometric difference of two or more angles. Some of the trigonometric identities regarding the compound angles are listed below:

  • sin(A + B) = sinA cosB + cosA sinB
  • sin(A – B) = sinA cosB – cos A sin B
  • cos(A + B) = cosA cosB – sinA sinB

Submultiples of an angle:

As the name suggests, trigonometric ratio of a submultiple of an angle means when we try to find out some trigonometric value of an angle of the type A/2 or A/3.

Some of the trigonometric identities for the submultiple of an angle are:

  • | sin A/2 + cos A/2| = √(1 + sin A)
  • | sin A/2 – cos A/2| = √(1 – sin A)
  • tan A/2 = ±√(1 – cos A)/(1 + cos A)

Negative Angles:

By convention, an angle measured in the anti-clockwise direction of positive x-axis is considered to be positive and hence, the angle measured in the clockwise direction is considered to be negative. Angle is an extremely important concept in trigonometry and the trigonometric results tend to change for negative angles.