Trigonometric Ratios of Compound Angles

TRIGONOMETRIC RATIOS OF COMPOUND ANGLES

3.1 Trigonometric Ratios for the Sum and Difference of Angles

Definitions and Equations

  • Sine of the Sum of Angles:
      The sine of the sum of angles A and B is given by the equation:
      extsin(A+B)=extsinAimesextcosB+extcosAimesextsinBext{sin}(A + B) = ext{sin}A imes ext{cos}B + ext{cos}A imes ext{sin}B

  • Sine of the Difference of Angles:
      The sine of the difference between angles A and B is given by the equation:
      extsin(A−B)=extsinAimesextcosB−extcosAimesextsinBext{sin}(A - B) = ext{sin}A imes ext{cos}B - ext{cos}A imes ext{sin}B

  • Cosine of the Sum of Angles:
      The cosine of the sum of angles A and B is given by the equation:
      extcos(A+B)=extcosAimesextcosB−extsinAimesextsinBext{cos}(A + B) = ext{cos}A imes ext{cos}B - ext{sin}A imes ext{sin}B

  • Cosine of the Difference of Angles:
      The cosine of the difference between angles A and B is given by the equation:
      extcos(A−B)=extcosAimesextcosB+extsinAimesextsinBext{cos}(A - B) = ext{cos}A imes ext{cos}B + ext{sin}A imes ext{sin}B

Tangent of Compound Angles

  • Tangent of the Sum of Angles:
      The tangent of the sum of angles A and B is defined by the equation:
      exttan(A+B)=racexttanA+exttanB1−exttanAimesexttanBext{tan}(A + B) = rac{ ext{tan}A + ext{tan}B}{1 - ext{tan}A imes ext{tan}B}

  • Tangent of the Difference of Angles:
      The tangent of the difference between angles A and B is given by the equation:
      exttan(A−B)=racexttanA−exttanB1+exttanAimesexttanBext{tan}(A - B) = rac{ ext{tan}A - ext{tan}B}{1 + ext{tan}A imes ext{tan}B}

Cotangent of Compound Angles

  • Cotangent of the Sum of Angles:
      The cotangent of the sum of angles A and B is given by:
      extcot(A+B)=racextcotA+extcotBextcotAimesextcotB−1ext{cot}(A + B) = rac{ ext{cot}A + ext{cot}B}{ ext{cot}A imes ext{cot}B - 1}

  • Cotangent of the Difference of Angles:
      The cotangent of the difference between angles A and B is expressed by:
      extcot(A−B)=racextcotAimesextcotB+1extcotB−extcotAext{cot}(A - B) = rac{ ext{cot}A imes ext{cot}B + 1}{ ext{cot}B - ext{cot}A}

Angles in Specific Ranges

  • The equations hold true for angles A and B defined by the forms:
      - A=nracextπ2A = n rac{ ext{π}}{2}
      - B=nd+racextπ2B = n d + rac{ ext{π}}{2} and similar conditions that ensure the appropriateness of their domains.

Tangent and Cotangent Summary for Three Angles

  • Tangent of Three Angles:
      The tangent of the sum of three angles A, B, and C is defined by:
      exttan(A+B+C)=racexttanA+exttanB+exttanC−exttanAimesexttanBimesexttanC1−exttanAimesexttanB−exttanBimesexttanC−exttanCimesexttanAext{tan}(A + B + C) = rac{ ext{tan}A + ext{tan}B + ext{tan}C - ext{tan}A imes ext{tan}B imes ext{tan}C}{1 - ext{tan}A imes ext{tan}B - ext{tan}B imes ext{tan}C - ext{tan}C imes ext{tan}A}

  • Cotangent of Three Angles:
      The cotangent of the sum of three angles A, B, and C is given by:
      extcot(A+B+C)=racextcotA+extcotB+extcotC−extcotAimesextcotBimesextcotC1−extcotAimesextcotB−extcotBimesextcotC−extcotCimesextcotAext{cot}(A + B + C) = rac{ ext{cot}A + ext{cot}B + ext{cot}C - ext{cot}A imes ext{cot}B imes ext{cot}C}{1 - ext{cot}A imes ext{cot}B - ext{cot}B imes ext{cot}C - ext{cot}C imes ext{cot}A}

Sine and Cosine of Three Angles

  • Sine of Three Angles:
      The sine of the sum of angles A, B, and C can be expressed as:
      extsin(A+B+C)=extsinAimesextcosBimesextcosC+extcosAimesextsinBimesextcosC+extcosAimesextcosBimesextsinCext{sin}(A + B + C) = ext{sin}A imes ext{cos}B imes ext{cos}C + ext{cos}A imes ext{sin}B imes ext{cos}C + ext{cos}A imes ext{cos}B imes ext{sin}C

  • Cosine of Three Angles:
      The cosine of the sum of angles A, B, and C is given by:
      extcos(A+B+C)=extcosAimesextcosBimesextcosC−extsinAimesextsinBimesextcosC−extsinAimesextcosBimesextsinC−extcosAimesextsinBimesextsinCext{cos}(A + B + C) = ext{cos}A imes ext{cos}B imes ext{cos}C - ext{sin}A imes ext{sin}B imes ext{cos}C - ext{sin}A imes ext{cos}B imes ext{sin}C - ext{cos}A imes ext{sin}B imes ext{sin}C

Product-to-Sum Formulas

  • Sine Product:
      The product of the sine of sums can be simplified as:
      extsin(A+B)imesextsin(A−B)=extsin2A−extsin2Bext{sin}(A + B) imes ext{sin}(A - B) = ext{sin}^2A - ext{sin}^2B

  • Cosine Product:
      The product of the cosine of sums can be simplified as:
      extcos(A+B)imesextcos(A−B)=extcos2A−extsin2Bext{cos}(A + B) imes ext{cos}(A - B) = ext{cos}^2A - ext{sin}^2B

This section elaborates on the formulas and definitions critical for working with compound angles in trigonometry, providing a solid framework for understanding and applying trigonometric identities in various mathematical contexts, particularly for IIT JEE preparation.