TRIGNOMETIC IDENTITIES

                 

Trigonometric Formulas: Trigonometry Formulas For Class 10, 11 & 12

Before getting into the trigonometric formula list, let us consider the following right-angled triangle: 

As you can see, the three sides of the triangle are:

a. Base: The side that is horizontal to the plane.

b. Perpendicular: The side making an angle of 90 degree with the Base.

c. Hypotenuse: The longest side of the triangle.

Also, $\theta$ is the angle made by Hypotenuse and Base.

Then,

sine of angle $\theta$ = $\sin \theta$ = $\frac{Perpendicular}{Hypotenuse}$

cosine of angle $\theta$ = $\cos \theta$ = $\frac{Base}{Hypotenuse}$

tangent of angle $\theta$ = $\tan \theta$ = $\frac{Perpendicular}{Base}$

cotangent of angle $\theta$ = $\cot \theta$ = $\frac{Base}{Perpendicular}$

cosecant of angle $\theta$ = $cosec\theta$ = $\frac{Hypotenuse}{Perpendicular}$

secant of angle $\theta$ = $\sec \theta$ = $\frac{Hypotenuse}{Base}$

Note that, sine, cosine, tangent, cotangent, cosecant, and secant are called Trigonometric Functions that defines the relationship between the sides and angles of the triangle.

Reciprocal Relationship Between Trigonometric Functions

The reciprocal relationship between different Trigonometric Functions are as under: $\tan \theta$ = $\frac{1}{\cot \theta }$ = $\frac{\sin \theta }{\cos \theta }$ $\cot \theta$ = $\frac{1}{\tan \theta }$ = $\frac{\cos \theta }{\sin \theta }$ $cosec\theta$ = $\frac{1}{\sin \theta }$ $\sec \theta$ = $\frac{1}{\cos \theta }$

Trigonometric Ratios Of Complementary Angles

First Quadrant

sin(π/2−$\theta$) = $\cos \theta$ cos(π/2−$\theta$) = $\sin \theta$ tan(π/2−$\theta$) = $\cot \theta$ cot(π/2−$\theta$) = $\tan \theta$ sec(π/2−$\theta$) = cosec$\theta$ cosec(π/2−$\theta$) = $\sec \theta$

Second Quadrant

sin(π−$\theta$) = $\sin \theta$ cos(π−$\theta$) = -$\cos \theta$ tan(π−$\theta$) = -$\tan \theta$ cot(π−$\theta$) = -$\cot \theta$ sec(π−$\theta$) = -sec$\theta$ cosec(π−$\theta$) = cosec$\theta$

Third Quadrant

sin(π+$\theta$) = -$\sin \theta$ cos(π+$\theta$) = -$\cos \theta$ tan(π+$\theta$) = $\tan \theta$ cot(π+$\theta$) = $\cot \theta$ sec(π+$\theta$) = -sec$\theta$ cosec(π+$\theta$) = -cosec$\theta$

Fourth Quadrant

sin(2π−$\theta$) = -$\sin \theta$ cos(2π−$\theta$) = $\cos \theta$ tan(2π−$\theta$) = -$\tan \theta$ cot(2π−$\theta$) = -$\cot \theta$ sec(2π−$\theta$) = sec$\theta$ cosec(2π−$\theta$) = -cosec$\theta$

Periodicity Identities

sin(2nπ + $\theta$) = $\sin \theta$ cos(2nπ + $\theta$) = $\cos \theta$ tan(2nπ + $\theta$) = $\tan \theta$ cot(2nπ + $\theta$) = $\cot \theta$ sec(2nπ + $\theta$) = $\sec \theta$ cosec(2nπ + $\theta$) = cosec$\theta$

Trigonometry Table

Trigonometry table is a table that you can refer to for the values of trigonometric ratios of different angles. Below is the table for trigonometry formulas of different angles which are commonly used for solving various problems.

Angles (In Degrees)	0°	30°	45°	60°	90°	180°	270°	360°
Angles (In Radians)	0°	π/6	π/4	π/3	π/2	π	3π/2	2π
sin	0	1/2	1/√2	√3/2	1	0	-1	0
cos	1	√3/2	1/√2	1/2	0	-1	0	1
tan	0	1/√3	1	√3	∞	0	∞	0
cot	∞	√3	1	1/√3	0	∞	0	∞
cosec	∞	2	√2	2/√3	1	∞	-1	∞
sec	1	2/√3	√2	2	∞	-1	∞	1

       

Trigonometric Identities

${\sin }^2\theta +{\cos }^2\theta =1$ ${\tan }^2\theta +1={\sec }^2\theta$ ${\cot }^2\theta +1=cose{c}^2\theta$

Sign Of Trigonometric Functions

$\sin (-\theta )=-\sin \theta$ $\cos (-\theta )=\cos \theta$ $\tan (-\theta )=-\tan \theta$ $cosec(-\theta )=-cosec\theta$ $\sec (-\theta )=\sec \theta$ $\cot (-\theta )=-\cot \theta$

Trigonometric Functions Of Sum And Difference Of Two Angles

$\sin (A+B)=\sin A\cos B+\cos A\sin B$ $\sin (A-B)=\sin A\cos B–\cos A\sin B$ $\cos (A+B)=\cos A\cos B–\sin A\sin B$ $\cos (A–B)=\cos A\cos B+\sin A\sin B$ $\tan (A+B)=\frac{\tan A+\tan B}{1–\tan A\tan B}$ $\tan (A–B)=\frac{\tan A–\tan B}{1+\tan A\tan B}$

Trigonometry Formulas Involving Product Identities

$\sin AsinB=\frac{1}{2}[\cos (A–B)-\cos (A+B)]$ $\cos A\cos B=\frac{1}{2}[\cos (A–B)+\cos (A+B)]$ $\sin A\cos B=\frac{1}{2}[\sin (A+B)+\sin (A-B)]$ $\cos A\sin B=\frac{1}{2}[\sin (A+B)–\sin (A-B)]$

Trigonometry Formulas Involving Sum To Product Identities

$\sin A+\sin B=2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)$ $\sin A-\sin B=2\cos \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right)$ $\cos A+\cos B=2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)$ $\cos A-\cos B=–2\sin \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right)$

Trigonometry Formulas Involving Double Angle Identities

$\sin 2A=2\sin A\cos A=\frac{2\tan A}{1+{\tan }^{2}A}$ $\cos 2A={\cos }^{2}A–{\sin }^{2}A=1–2si{n}^{2}A=2co{s}^{2}A–1=\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}$ $\tan 2A=\frac{2\tan A}{1–{\tan }^{2}A}$

Trigonometry Formulas Involving Triple Angle Identities

$\sin 3A=3\sin A–4{\sin }^{3}A=4\sin ({60}^{\circ }-A).\sin A.\sin ({60}^{\circ }+A)$ $\cos 3A=4{\cos }^{3}A–3\cos A=4\cos ({60}^{\circ }-A).\cos A.\cos ({60}^{\circ }+A)$ $\tan 3A=\frac{3\tan A–{\tan }^{3}A}{1-3{\tan }^{2}A}=\tan ({60}^{\circ }-A).\tan A.\tan ({60}^{\circ }+A)$

Trigonometry Formulas Involving Half Angle Identities

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

Trigonometry Formulas: Inverse Properties

$\theta ={\sin }^{-1}\left(x\right)isequivalenttox=\sin \theta$ $\theta ={\cos }^{-1}\left(x\right)isequivalenttox=\cos \theta$ $\theta ={\tan }^{-1}\left(x\right)isequivalenttox=\tan \theta$ $\sin \left({\sin }^{-1}\left(x\right)\right)=x$ $\cos \left({\cos }^{-1}\left(x\right)\right)=x$ $\tan \left({\tan }^{-1}\left(x\right)\right)=x$ ${\sin }^{-1}\left(\sin \left(\theta \right)\right)=\theta$ ${\cos }^{-1}\left(\cos \left(\theta \right)\right)=\theta$ ${\tan }^{-1}\left(\tan \left(\theta \right)\right)=\theta$

Given below are some more inverse trigonometry formulas

sin -1 (-x) = – sin -1 x cos -1 (-x) = – sin -1 x tan -1 (-x) = – tan -1 x cosec -1 (-x) = – cosec -1 x sec -1 (-x) = – sec -1 x cot -1 (-x) = – cot -1 x

sin -1 (1/x) = cosec -1 x cos -1 (1/x) = sec -1 x tan -1 (1/x) = cot -1 x tan -1 (1/x) = cot -1 x

sin -1 (x) + cos -1 (x) = π/2 tan -1 (x) + cot -1 (x) = π/2 sec -1 (x) + cosec -1 (x) = π/2

Inverse Trigonometry Substitution

Expression	Substitution	Identity
√a2 − x2	x = a sin θ	1 – sin2 θ = cos2 θ
√a2 + x2	x = a tan θ	1 – tan2 θ = sec2 θ
√x2 − a2	x = a sec θ	sec2 θ – 1 = tan2 θ