Multi-Angle Identities

In this section, we will study, relations by which trigonometric ratios of more than one angle are governed. Since, this course is for competitive exams we are not discussing the proof, how these identities have been achieved. However, you may verify the identities with angle 0°, 30°, 45°, 60° & 90°.

We will discuss the following Identities:-

1) Sum of two Trigonometric ratios of different angles like sin A + sin B

2) Product of two Trigonometric ratios of different angles like sin A X sin B

3) Sum of two Angles like sin (A+B) & its derivative when A = B, the concept for double angle like sin2θ and half angle.

4) Sum of three equal Angles like sin3θ

♦ \(sinA+sinB=2\times sin({A+ B\over2})\times cos({A- B\over2})\)

Let us assume A = 60° and B = 30°

\(sin60^0+sin30^0=2\times sin({60+ 30\over2})\times cos({60-30\over2})\)

\({\sqrt3\over 2}+{1\over 2}=2\times {1\over\sqrt 2}\times {{\sqrt3+1}\over 2\sqrt2}\)

\({{\sqrt3+1}\over 2} ={{\sqrt3+1}\over 2}\) [Hence, we can see that both the equations are equal]

Similarly,

♦ \(sinA-sinB=2\times sin({A- B\over2})\times cos({A+ B\over2})\)

♦ \(cosA+cosB=2\times cos({A+ B\over2})\times cos({A- B\over2})\)

♦ \(cosA-cosB=-2\times sin({A+ B\over2})\times sin({A- B\over2})\)

♦ \(sinA\times sinB={1\over2}[cos(A- B)-cos(A+ B)]\)

Let us assume A = 60° and B = 30°

\(sin60^0\times sin30^0={1\over2}[cos(60-30)-cos(60+30)]\)

\({\sqrt3\over 2}\times{1\over 2}={1\over2}[{\sqrt3\over 2}-0]\)

\({\sqrt3\over 4}={\sqrt3\over 4}\) [Hence, we can see that both the equations are equal]

Similarly,

♦ \(cosA\times cosB={1\over2}[cos(A- B)+cos(A+ B)]\)

♦ \(sinA\times cosB={1\over2}[sin(A+ B)+sin(A- B)]\)

♦ \(cosA\times sinB={1\over2}[sin(A+ B)-sin(A- B)]\)

♦ \(sin\space (A+B)= sin A\space cos B \space +\space cosA\space sin B\)

Let us assume A = 60° and B = 30°
\(sin (60+30)= sin 60^0\space cos 30^0 +\space cos60^0 sin 30^0\)

1 = \({\sqrt3\over 2} \times {\sqrt3\over 2}+{1\over 2} \times {1\over 2} \)

1 = 1 [Hence, we can see that both the equations are equal]

Similarly,

♦ \(sin\space (A-B)= sin A\space cos B \space -\space cosA\space sin B\)

♦ \(cos\space (A+B)= cos A\space cos B \space -\space sinA\space sin B\)

♦ \(cos\space (A-B)= cos A\space cos B \space +\space sinA\space sin B\)

♦ \(tan\space (A+B)= {tan A\space +\space tan B\over 1\space -\space tanA\space tanB}\)

♦ \(tan\space (A-B)= {tan A\space -\space tan B\over 1\space +\space tanA\space tanB}\)

Double Angle Formulas

♦ \(sin\space 2\theta = 2\space sin\theta \space cos \theta\)

♦ \(cos\space 2\theta = cos^2\theta-sin^2\theta\)

♦ \(cos\space 2\theta = 1- 2sin^2\theta\)

♦ \(cos\space 2\theta = 2\space cos^2\theta-1\)

♦ \(tan\space 2\theta= {2\space tan\theta\over1-tan^2\theta}\)

Note: The formulas for double angle are achieved when A = B = θ, however, we should learn both the concepts given above. 

Half Angle Formulas

The concept of \(cos\space 2\theta = cos^2\theta-sin^2\theta\) is used to achieve half-angle formulas, by this approach a given angle can be doubled like

♦ \(sin\space {\theta\over2}=\pm\sqrt{1-cos\theta\over2}\)

♦ \(cos\space {\theta\over2}=\pm\sqrt{1+cos\theta\over2}\)

♦ \(tan\space {\theta\over2}=\pm\sqrt{1-cos\theta\over 1+ cos\theta}={1-cos\theta\over sin\theta} = {sin\theta\over 1+cos\theta}\)

Note: You may learn or just understand the concept behind it. These formulas can also be termed as power reducing formulas.

♦ \(sin\space 3\theta = 3\space sin\theta-4\space sin^3\theta\)

♦ \(cos\space 3\theta = 4\space cos^3\theta-3\space cos\theta\)

♦ \(tan\space 3\theta={3\space tan\theta-tan^3\theta\over 1- 3\space tan^2\theta}\)

Note: derivation of the above formula is not important, however, it is recommended to learn the formulas as many times we require them in solving questions based on trigonometric equations.