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Trigonometry Formulas & Identities – Table, Simplest Way to Learn, Complete List

Trigonometry Formula

Trigonometry is considered as the branch of mathematics that involves relation with the sides and angles of a triangle. There are various types of problems in trigonometry that can be solved with the help of Trigonometry Formula. Along with these Trigonometry Formulas and Identities, trigonometry table is also helpful in finding the solution of the given problems.

Here, on this page we have provided complete list of Trigonometry Formulas & Identities. Also, the simplest ways to learn the Trigonometry Table in step wise procedure is mentioned in the below segment. So, candidates must go through this complete page to gain knowledge regarding the Trigonometry Topic.

Trigonometry Formula

Trigonometry Formulas List:

Here, we have list of formulas for trigonometry. The formulas of trigonometry are based on right-angled triangles only which includes three sides – Hypotenuse, Opposite side (Perpendicular), and Adjacent side (Height).

Following is the right angled triangle and the three sides are as:

  • Base: The side that is horizontal to the plane.
  • Perpendicular: The side making an angle of 90 degree with the Base.
  • Hypotenuse: The longest side of the triangle.

Also, θ is the angle made by Hypotenuse and Base.

These are the list of Trigonometry Formulas:

  • Basic Formulas
  • Reciprocal Identities
  • Trigonometry Table
  • Periodic Identities
  • Co-function Identities
  • Sum and Difference Identities
  • Double Angle Identities
  • Triple Angle Identities
  • Half Angle Identities
  • Product Identities
  • Sum to Product Identities
  • Inverse Trigonometry Formulas

Also, Get Here: Trigonometry Short Tricks

Trigonometry Formulas – in detailed way:

1) Basic Formulas:

6 ratios are used in trigonometry and these are called as trigonometric functions. The names of six ratios are sine, cosine, secant, co-secant, tangent and co-tangent. In these six ratios, we use right angled triangle as a reference.

  • sin θ = Opposite Side/Hypotenuse
  • cos θ = Adjacent Side/Hypotenuse
  • tan θ = Opposite Side/Adjacent Side
  • sec θ = Hypotenuse/Adjacent Side
  • cosec θ = Hypotenuse/Opposite Side
  • cot θ = Adjacent Side/Opposite Side

2) Reciprocal Identities:

The Reciprocal Identities are as:

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

3) Trigonometry Table:

The below given is table for trigonometry formulas for angles by using which we solve our problems:

Angles (In Degrees)30°45°60°90°180°270°360°
Angles (In Radians)π/6π/4π/3π/2π3π/2

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4) Periodicity Identities (in Radians):

These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called co-function identities.

  • sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
  • sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
  • sin (3π/2 – A)  = – cos A & cos (3π/2 – A)  = – sin A
  • sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
  • sin (π – A) = sin A &  cos (π – A) = – cos A
  • sin (π + A) = – sin A & cos (π + A) = – cos A
  • sin (2π – A) = – sin A & cos (2π – A) = cos A
  • sin (2π + A) = sin A & cos (2π + A) = cos A

5) Co-function Identities (in Degrees):

The co-function or periodic identities can also be represented in degrees as:

  • sin(90°−x) = cos x
  • cos(90°−x) = sin x
  • tan(90°−x) = cot x
  • cot(90°−x) = tan x
  • sec(90°−x) = csc x
  • csc(90°−x) = sec x

6) Sum & Difference Identities:

  • sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
  • cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
  • tan(x+y) = (tan x + tan y)/ (1−tan x . tan y)
  • sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
  • cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
  • tan(x−y) = (tan x–tan y)/ (1+tan x . tan y)

Get Here: Simplification and Approximation Questions

7) Double Angle Identities:

sin(2x) = 2sin(x) . cos(x) = [2tan x/(1+tan2 x)]

cos(2x) = cos2(x)–sin2(x) = [(1-tanx)/(1+tan2 x)]

cos(2x) = 2cos2(x)−1 = 1–2sin2(x)

tan(2x) = [2tan(x)]/ [1−tan2(x)]

sec (2x) = sec2 x/(2-sec2 x)

csc (2x) = (sec x. csc x)/2

8) Triple Angle Identities:

  • Sin 3x = 3sin x – 4sin3x
  • Cos 3x = 4cos3x-3cos x
  • Tan 3x = [3tanx-tan3x]/[1-3tan2x]

9) Half Angle Identities:

Solve Here: Chain Rule Aptitude Questions and Answers

10) Product identities:

11) Sum to Product Identities:

12) Inverse Trigonometry Formulas:

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

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Simplest Way to Learn Trigonometry Table:

If you know the trigonometry formulas, then you can easily remember the trigonometry table.

Firstly, try to learn the below given trigonometry formulas:

  • sin x = cos (90° – x)
  • cos x = sin (90° – x)
  • tan x = cot (90° – x)
  • cot x = tan (90° – x)
  • sec x = cosec (90° – x)
  • cosec x = sec (90° – x)
  • 1/sin x = cosec x
  • 1/cos x = sec x
  • 1/tan x = cot x

Steps to create Trigonometry Table:

Step 1) Create a table and on the top row, write the angles such as 0°, 30°, 45°, 60°, 90° and on the first column, write the trigonometric functions such as sin, cos, tan, cosec, sec, cot.

Step 2) Find the value of sin.

For writing the values of sin, you can divide 0, 1, 2, 3, 4 by 4 under the root, such as the value of sin 0°=

Angles (In Degrees)30°45°60°90°180°270°360°

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Step 3) Find the value of cos.

The value of cos is the opposite angle of the sin angle. For cos value, divide 4 by 4 under the root to get the value of cos 0°.

To determine the value of cos 0° =

Angles (In Degrees)30°45°60°90°180°270°360°

Step 4) Find the value of tan.

tan = sin/cos. For the value of tan at 0°, divide the value of sin at 0° by the value of cos at 0°.

tan 0°= 0/1 = 0

Angles (In Degrees)30°45°60°90°180°270°360°

Step 5) Find the value of cot.

The value of cot is equal to the reciprocal of tan. The value of cot at 0° will be obtained by dividing 1 by the value of tan at 0°.

cot 0° = 1/0 = Infinite or Not Defined

Angles (In Degrees)30°45°60°90°180°270°360°

Step 6) Find the value of cosec.

The value of cosec at 0° is the reciprocal of sin at 0°.

cosec 0°= 1/0 = Infinite or Not Defined

Angles (In Degrees)30°45°60°90°180°270°360°

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Step 7) Find the value of sec.

The value of sec is equal to all the reciprocal values of cos. The value of sec on 0° is the opposite of cos on 0°.

sec0°= 1/1=1

Angles (In Degrees)30°45°60°90°180°270°360°

We the team of has covered up complete list of Trigonometry Formulas & Identities along with simplest ways to remember the Trigonometry Table. Students must go through it, while preparing for the Trigonometry Chapter.

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