Trigonometry

Trigonometry - Formulas, Identities, Functions and Problems

Trigonometry is the branch of mathematics which is basically concerned with specific functions of angles, their applications and their calculations. In mathematics, there are a total of six different types of trigonometric functions: Sine (sin), Cosine (cos), Secant (sec), Cosecant (cosec), Tangent (tan) and Cotangent (cot). These six different types of trigonometric functions symbolize the relation between the ratios of different sides of a right angle triangle. These trigonometric functions can also be called as circular functions as their values can be described as the ratios of x and y co-ordinates of the circle of radius 1 that keep in touch with the angles in standard positions.

The relation between these trigonometric identities with the sides of the triangles can be given as follows:-


  • • Sine (theta) = Opposite/Hypotenuse

  • • Cos (theta)= Adjacent/Hypotenuse

  • • Tan (theta)= Opposite/Adjacent

  • • Cot (theta)= Adjacent/Opposite

  • • Cosec (theta)= Hypotenuse/Opposite

  • • Sec (theta)= Hypotenuse/Adjacent


  • The trigonometric functions are very important for studying triangles, light, sound or wave. The values of these trigonometric functions in different domains and ranges can be used from the following table:

    Trigonometric functionsDomainRange
    Sin xR-1 ≤ sin x ≤ 1
    Cos xR-1 ≤ cos x ≤ 1
    Tan xR – {(2n + 1)π/2, n ∈ I}R
    Cosec xR – {nπ, n ∈ I}R – {x: -1 < x < 1}
    Sec xR – {(2n + 1)π/2, n ∈ I}R – {x: -1 < x < 1}
    Cot xR – {nπ, n ∈ I}R



    The values of different trigonometric functions at different angles are given in the following table through which it can be directly used in the problems:

    Angles30°45°60°90°
    sin01/21/√2√3/21
    cos1√3/21/√21/20
    tan0√3/21√3undefined
    cosecundefined2√22/√31
    sec12/√3√22undefined
    cotundefined√31√3/20


    Some common Identities and formulas generally used in finding Trigonometric ratios are stated below:


    Double or Triple angle identities:-

       1) sin 2x = 2sin x cos x
       2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1
       3) tan 2x = 2 tan x / (1-tan 2x)
       4) sin 3x = 3 sin x – 4 sin3x
       5) cos3x = 4 cos3x – 3 cosx
       6) tan 3x = (3 tan x – tan3x) / (1- 3tan 2x)

    Sum and difference formulas of different trigonometric functions are as follows:


       1) sin (a + ß) = sin(a) cos(ß) + cos(a) sin(ß)
       2) sin (a – ß) = sin(a) cos(ß) – cos(a) sin(ß)
       3) cos (a + ß) = cos(a) cos(ß) – sin(a) sin(ß)
       4) cos (a – ß) = cos(a) cos(ß) + sin(a) sin(ß)
       5) tan (a + ß) = [tan(a) + tan (ß)]/ [1 – tan(a) tan(ß)]
       6) tan (a – ß) = [tan(a) – tan (ß)]/ [1 + tan (a) tan (ß)]
       7) tan (π/4 + θ) = (1 + tan θ)/(1 – tan θ)
       8) tan (π/4 – θ) = (1 – tan θ)/(1 + tan θ)
       9) cot (a + ß) = [cot(a) . cot (ß) – 1]/ [cot (a) + cot (ß)]
       10) cot (a – ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) – cot (a)]

    For triple angle, the below mentioned trigonometric functions are used:


      1) 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
       2) 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
       3) tan (A + B + C) = [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
       4) cot (A + B + C) = [cot A cot B cot C – cot A – cot B – cot C]/ [cot A cot B + cot B cot C + cot A cot C – 1]

    The relations between different trigonometric functions are as follows:-


  • 1) Sin A = 1/cosec A

  • 2) Cos A = 1/sec A

  • 3) Sec A = 1/cos A

  • 4) Cosec A = 1/sin A

  • 5) Tan A = 1/cot A, sin A/cos A

  • 6) Cot A = 1/tan A, cos A/sin A

  • For periodicity identities between trigonometry functions:-


  • 1) Sin (x + 2π ) = Sinx

  • 2) Cos (x + 2π ) = Cosx

  • 3) Tan (x + π ) = Tanx

  • 4) Cot (x + π ) = Cotx

  • For half angle trigonometry functions:-


  • 1) Sin x/2= ± √(1-cos x)/2

  • 2) Cos x/2= ± √(1+cos x)/2

  • 3) Tan x/2= √(1- cos x)/ (1+ cos x) , (1- cos x)/ sin x

  • For sum to product trigonometric identities:-


  • 1) \[Sin\:\alpha\pm Sin\:\beta=2Sin\frac{1}{2}\:(\alpha \pm \beta ) \:Cos\frac{1}{2}\:(\alpha \mp \beta )\]

  • 2) \[Cos\:\alpha+ Cos\:\beta=2Cos\frac{1}{2}\:(\alpha + \beta ) \:Cos\frac{1}{2}\:(\alpha - \beta )\]

  • 3) Cos α – Cos β = - 2 Sin [(α + β)/ 2 ] sin [(α – β)/ 2]

  • Square law formulas:-


  • 1) Sin2x + cos2x = 1

  • 2) Tan2x = 1+ sec2x

  • 3) Cot2x = 1+cosec2x

  • Along with the knowledge that the two acute angles are complimentary i.e. they add to 90° and you can solve any right triangle:

  • • If you know two of the three sides, you can find the third side and both the acute angles.

  • • If you know one acute angle and one of the three sides, you can find the other acute angle and the other two sides.

  • Signs of trigonometric functions play an important role in their formulas, as sign changes with the change in a quadrant. Basically, the sign is based on the quadrant in which the angle lies.

  • • In Q1, all trigonometric ratios are positive. (Angles between 0° – 90°)

  • • In Q2, all trigonometric ratios of sinθ and cosecθ are positive. (Angles between 90° – 180°)

  • • In Q3, all trigonometric ratios of cosθ and secθ are positive. (Angles between 180° – 270°)

  • • In Q4, all trigonometric ratios of tanθ and cotθ are positive. (Angles between 270° – 360°)

  • The values of trigonometric functions will change with the change in the angles, but the value remains the same for 90o ± θ and 270o ± θ and that of 180o ± θ and 360o ± θ, as we add or subtract θ from 90o ± θ and 270o ± θ we get,

  • • Sec (90o + θ ) = Cos θ

  • • Cot (90o – θ ) = Cos θ

  • • Tan (90o + θ ) = – Cot  θ

  • • Tan (90o – θ ) = Cot  θ

  • • Sec (90o + θ ) = Cosec θ

  • • Sec (90o + θ ) = Cosec θ

  • • Sin (270o – θ ) = – Cos θ

  • • Sin (270o – θ ) = – Cos θ

  • Some important inverse trigonometry formulas can easily be remembered through the following domain and range of the inverse trigonometric identities:

    Trigonometric functionsDomainRange
    Sin-1 x[-1,1][-π/2, π/2]
    Cos-1 x[-1,1][0, π]
    Tan-1 xR[-π/2, π/2]
    Cot-1 xR[0, π]
    Sec-1 xR - (-1,1)[0, π] – [ π/2 ]
    Cosec-1 xR - (-1,1)[-π/2, π/2] – [0]


    The above table of domain and range of the trigonometric identities shows that the Sin-1 x has infinitely many solutions at x € [-1, 1], and there is only a single value which lies in the intervals [π/2, π/2], which is termed as the principal value.