Trigonometry All Formulas and Identities – JEE Mains, Advanced

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Trigonometry All Formulas and Identities, complete list available here. Useful for JEE Mains, Advanced, and other state government examinations.

Trigonometry All Formulas

Different types of trigonometry formulas need to be remembered to solve the problems. Check the Trigonometric Ratios (sin, cos, tan, sec, cosec & cot), Sign of the Trigonometric Functions, Trigonometric Functions, Transformation Formulae, Multiple and Sub-Multiple Angles, Conditional Identities, etc below.

Must remember these formulas for fast calculations, especially for IIT JEE aspirants.

Basic Trigonometric Formulas

  • sin θ = Perpendicular (Opposite Side)/Hypotenuse
  • cos θ = Base (Adjacent Side)/Hypotenuse
  • tan θ = Perpendicular (Opposite Side)/Base (Adjacent Side)
  • cosec θ = Hypotenuse/Perpendicular (Opposite Side)
  • sec θ = Hypotenuse/Base (Adjacent Side)
  • cot θ = Base (Adjacent Side)/Perpendicular

Reciprocal Identities

  • sin θ = 1/cosec θ
  • cos θ = 1/sec θ
  • tan θ = 1/cot θ
  • cosec θ = 1/sin θ
  • sec θ = 1/cos θ
  • cot θ = 1/tan θ

Trigonometric Formulas Table

Angles(In Degrees)0°30°45°60°90°180°270°360°
Angles(In Radians)0π/6π/4π/3π/2π3π/22π
Sin01/21/√2√3/210-10
Cos1√3/21/√21/20-101
Tan01/√31√3∞0∞0
Cot∞√311/√30∞0∞
Cosec∞2√22/√31∞-1∞
Sec12/√3√22∞-1∞1

Cofunction Identities

I – Quadrant

  • sin(π/2 − θ) = cos θ
  • cos(π/2 − θ) = sin θ
  • tan(π/2 − θ) = cot θ
  • cot(π/2 − θ) = tan θ
  • sec(π/2 − θ) = cosec θ
  • cosec(π/2 − θ) = sec θ
Related Question:  If tanA-tanB=x and cotB-cotA=y, then cot(A-B) is equal to

II – Quadrant

  • sin(π − θ) = sin θ
  • cos(π − θ) = -cos θ
  • tan(π − θ) = -tan θ
  • cot(π − θ) = – cot θ
  • sec(π − θ) = -sec θ
  • cosec(π − θ) = cosec θ

III – Quadrant

  • sin(π + θ) = – sin θ
  • cos(π + θ) = – cos θ
  • tan(π + θ) = tan θ
  • cot(π + θ) = cot θ
  • sec(π + θ) = -sec θ
  • cosec(π + θ) = -cosec θ

IV – Quadrant

  • sin(2π − θ) = – sin θ
  • cos(2π − θ) = cos θ
  • tan(2π − θ) = – tan θ
  • cot(2π − θ) = – cot θ
  • sec(2π − θ) = sec θ
  • cosec(2π − θ) = -cosec θ

Sum & Difference Identities (Formulas)

  • sin (A + B) = sinA cosB + cosA sinB
  • sin (A – B) = sinA cosB – cosA sinB
  • cos (A + B) = cosA cosB – sinA sinB
  • cos (A – B) = cosA cosB + sinA sinB
  • tan (A + B) = (tanA + tanB) / (1 – tanA tanB)
  • tan (A – B) = (tanA – tanB) / (1 + tanA tanB)
  • $cot (A+B) = \frac{\cot A\cot B\ -\ 1}{\cot A\ +\cot B}$
  • $cot (A-B)= \frac{\cot A\cot B\ +\ 1}{\cot A\ -\cot B}$
  • sin (A+B+C) = sin A cos B cos C + cos A sin B cos C + cos A cos B sin C – sin A sin B sin C
  • cos (A+B+C) = cos A cos B cos C- cos A sin B sin C – sin A cos B sin C – sin A sin B cos C
  • $tan (A+B+C) = \frac{(\tan A+\tan B+\tan C-\tan A\tan B\tan C)}{(1-\tan A\tan B-\tan B\tan C-\tan A\tan C)}$

Compound Angle Formulas

  • $\tan \ \left(\frac{\pi }{4}+\theta \right)\ =\ \frac{1+\tan \theta }{1-\tan \theta }$
  • $\tan \ \left(\frac{\pi }{4}+\theta \right)\ =\ \frac{\cos \theta \pm \sin \theta }{\cos \theta ∓\sin \theta }$
  • $\tan \ \left(\frac{\pi }{4}-\theta \right)\ =\ \frac{1-\tan \theta }{1+\tan \theta }=\ \frac{\cos 2\theta }{1+\sin 2\theta }$
  • $\sin (A+B)\sin (A-B)=\sin ^2A-\sin ^2B$
  • $\cos (A+B)\cos (A-B)=\cos ^2A-\sin ^2B\ =\ \cos ^2B-\sin ^2A$
  • If A = B+C, then tanA tanB tanC = tanA – tanB – tanC (A-> Greater Angle)
Related Question:  If cos A = 3/4,then the value of 16 cos²(A/2) −32sin(A/2)sin(5A/2) is

Double Angle Identities (Formulas)

  • $\sin 2A=2\sin A\cos A\ =\ \frac{(2\tan A)}{(1+\tan 2A)}$
  • $\cos 2\theta =\cos ^2\theta -\sin ^2\theta \ =\ 2\cos ^2\theta -1\ =\ 1-2\sin ^2\theta \ =\ \frac{(1-\tan ^2\theta )}{(1+\tan ^2\theta )}$
  • $\tan 2\theta =\ \frac{2\tan \theta }{(1-\tan ^2\theta )}$
  • $\sec 2\theta =\ \frac{\sec ^2\theta }{(2-\sec ^2\theta )}$
  • $\operatorname{cosec}2\theta =\ \frac{\sec \theta \ \operatorname{cosec}\theta }{2}$

Some Useful Formulas – Double Angle

  • $1+\cos 2\theta \ =\ 2\cos ^2\theta$
  • $1-\cos 2\theta \ =\ 2\sin ^2\theta$
  • $1\pm \sin 2\theta \ =\ \left(\cos \theta \pm \sin \theta \right)^2$
  • $\cot \theta -\tan \theta \ =\ 2\cot 2\theta$
  • $\cos \theta \ =\ \pm \sqrt{\frac{\left(1+\cos 2\theta \right)}{2}}$
  • $\sin \theta \ =\ \pm \sqrt{\frac{\left(1-\cos 2\theta \right)}{2}}$
  • $\tan \theta \ =\ \frac{\left(1-\cos 2\theta \right)}{\sin 2\theta }$
  • $\tan ^2\theta \ =\ \frac{\left(1-\cos 2\theta \right)}{\left(1+\cos 2\theta \right)}$

Triple Angle Identities (Formulas)

  • Sin 3θ = 3sin θ – 4sin3θ
  • Cos 3θ = 4cos3θ – 3cosθ
  • $\tan 3\theta \ =\ \frac{\left(3\tan \theta \ -\ \tan ^3\ \theta \right)}{1-3\tan ^2\theta }$

Half Angle Formulas

  • $\sin \theta =2\sin \frac{\theta }{2}\cos \frac{\theta }{2}$
  • $\cos \theta =\cos ^2\frac{\theta }{2}-\sin ^2\frac{\theta }{2}$
  • $\sin \frac{\theta }{2}\ =\ \pm \sqrt{\frac{\left(1-\cos \theta \right)}{2}\ }$
  • $\cos \frac{\theta }{2}\ =\ \pm \sqrt{\frac{\left(1+\cos \theta \right)}{2}}$
  • $\tan \frac{\theta }{2}\ =\ \frac{\left(1-\cos \theta \right)}{1+\cos \theta }\ =\ \frac{\left(1-\cos \theta \right)}{\sin \theta }$

Transformation Formulas

Sum to Product Identities

  • $\sin C+\sin D=2\sin \ \left(\frac{C+D}{2}\right)\ \cos \ \left(\frac{C-D}{2}\right)$
  • $\sin C-\sin D=2\ \cos \ \left(\frac{C+D}{2}\right)\ \sin \ \left(\frac{C-D}{2}\right)$
  • $\cos C+\cos D=2\ \cos \left(\frac{C+D}{2}\right)\ \cos \left(\frac{C-D}{2}\right)$
  • $\cos C-\cos D=-2\ \sin \left(\frac{C+D}{2}\right)\ \sin \left(\frac{C-D}{2}\right)=2\ \sin \left(\frac{C+D}{2}\right)\ \sin \left(\frac{D-C}{2}\right)$

Product Identities

  • $2\sin A\cos B=\sin (A+B)+\sin (A-B)$
  • $2\cos A\ \sin B=\sin (A+B)-\sin (A-B)$
  • $2\cos A\ \cos B=\cos (A+B)+\cos (A-B)$
  • $2\sin A\ \sin B=\cos (A-B)+\cos (A+B)$
Related Question:  If tanA-tanB=x and cotB-cotA=y, then cot(A-B) is equal to

Conditional Identities

$If\ A+B+C\ =\ \pi$

  • $\sin 2A+\sin 2B+\sin 2C\ =\ 4\sin A\sin B\sin C$
  • $\cos 2A+\cos 2B+\cos 2C\ =1-\ 4\cos A\cos B\cos C$
  • $\tan \left(A+B+C\right)\ =\frac{\left(\tan A+\tan B+\tan C\ -\tan A\tan B\tan C\right)}{1-\left(\tan A\tan B+\tan B\tan C+\tan C\tan A\right)}$
  • $\sin A+\sin B+\sin C\ =\ 4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$
  • $\cos A+\cos B+\cos C\ =\ 1+4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$
  • $\tan \frac{A}{2}\tan \frac{B}{2}+\tan \frac{B}{2}\tan \frac{C}{2}+\tan \frac{C}{2}\tan \frac{A}{2}=1$
  • $\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}=\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}$

Other Formulas for JEE IIT

Sine Cosine Series

Here, a ---> 1st angle, d ---> common difference, n ---> no.of terms
  • $\sin \ \left(a\right)\ +\ \sin \ \left(a+d\right)+\ \sin \ \left(a+2d\right)+…..+\sin \left(a+\left(n-1\right)d\right)\ =\ \frac{\sin \left(\frac{nd}{2}\right)}{\sin \left(\frac{d}{2}\right)}.\sin \ \left[a+\left(n-1\right)\frac{d}{2}\right]$
  • $\cos \left(a\right)\ +\cos \left(a+d\right)+\cos \left(a+2d\right)+…..+\cos \left(a+\left(n-1\right)d\right)\ =\ \frac{\sin \left(\frac{nd}{2}\right)}{\sin \left(\frac{d}{2}\right)}.\cos \ \left[a+\left(n-1\right)\frac{d}{2}\right]$

Product Series – Cosine Angles

  • $\cos \theta .\ \cos 2\theta .\ \cos 2^2\theta ………\cos 2^{n-1}\theta \ =\ \frac{\sin \left(2^n\theta \right)}{2^n\sin \theta }$

Range – Trigonometric Expression

$Expression=a\ \sin \theta \ +\ b\ \cos \theta$

  • $E=\ \sqrt{a^2+b^2}\left(\ \frac{a}{\sqrt{a^2+b^2}}\ \sin \theta \ +\ \frac{b}{\sqrt{a^2+b^2}}\cos \theta \right) $
  • $\therefore -\sqrt{a^2+b^2}\le a\ \sin \theta \ +\ b\ \cos \theta \le \sqrt{a^2+b^2}$

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