Trigonometry Formula : Trigonometry all Formulas and Identities List (Formula List)

Trigonometry Formula : Trigonometry all Formulas and Identities List (Formula List)

Trigonometry Basic Formula ;

This is the basic formula of trigonometry , here are the list of this

• 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

Reciprocal Identities of Trigonometry ;

Here are the some reciprocal identities of trigonometry ,

• 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

Pythagorean Identities of Trigonometry ;

Friends, these are Pythagorean Identities of trigonometry. It is based on Pythagoras theorem. This is a theorem based on the right angle trial in which the square of the hypotenuse is equal to the sum of the square of the perpendicular and the square of the base.

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

Ranges of the Trigonometric Functions;

The ranges of the trigonometry function sine, cosine, tangent, cotangent, secant and cosecant are as follows:

• −1 ≤ sin θ ≤ 1
• −1 ≤ cos θ ≤ 1
• −∞ ≤ tan θ ≤ ∞
• csc θ ≥ 1 and csc θ ≤ −1
• sec θ ≥ 1 sec θ ≤ −1
• −∞ ≤ cot θ ≤ ∞

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) = cosec x
• cosec(90°−x) = sec x

Periodicity Identities ;

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

Inverse Trigonometry Formulas;

Inverse trigonometry functions are also known as reverse trigonometry function;

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

Sum & Difference Identities;

here are the four sum and difference indentities formula of trigonometry ,

• sin(a+b)=sin(a)cos(b)+cos(a)sin(b)
• cos(a+b)=cos(a)cos(b)-sin(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)

Product to Sum Formulas;

• sin a sin b = 1/2 [cos (a-b) – cos (a+b)]
• cos a cos b = 1/2 [cos (a+b) + cos (a-b)]
• sin a cos b = 1/2 [sin (a+b) + sin (a-b)]
• cos a sin b = 1/2 [sin (a+b) – sin (a-b)]

Double Angle Formulas;

• sin (2θ) = 2sinθ cosθ = 2tanθ/1+tan²θ
• cos (2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ
• tan2θ = 2 tanθ / 1-tan²θ

Triple Angle Formulas;

• sin 3θ = 3sinθ – 4sin³θ
• cos 3θ = 4cos³θ – 3cosθ

Sine Law;

The Law of Sines relates the ratios of a triangle’s side lengths to the sines of its angles, stating that the sine of an angle divided by the length of the opposite side is constant for all angles in the triangle

sin α/a = sinβ/b = sin γ/c

Cosine Law;

The Law of Cosines is a trigonometric formula used to find the lengths of sides in non-right triangles, relating the square of a side to the sum of the squares of the other two sides minus twice the product of their lengths and the cosine of the included angle.

a² = b² + c² − 2bc cos α
b² = a² + c² − 2ac cos β
c² = a² + b² − 2ab cos γ