1. Trigonometry Formulas
Trigonometry is one of the most important yet difficult branches of
mathematics, and you will come across dozens of problems that will need
Trigonometry Formulas. But the question is, are you fully familiar with
Trigonometric formulas?
If not, then don’t worry. We have got you covered! In this post, we are
going to share a detailed and easy-to-understand list of all the
Trigonometry formulas, along with identities and functions. But before
that, let’s take a brief look at what Trigonometry is. Read on!
What is Trigonometry?
Trigonometry basically studies and deals with the relationship of lengths,
angles, and sides of triangles. Moreover, trigonometric functions and
2. formulas are widely used in navigating and surveying syst ems, geography,
and astronomy too!
The most common Trigonometric Functions you will hear are tan
(tangent), sin θ (sine), cos θ (cosine), sec θ (secant), cot θ (cotangent), and
cosec θ (cosecant). Don’t panic, we will explain these Trigonometric
Functions in detail, just try to stick till the end!
List of Trigonometry Formulas
As we have briefly mentioned above, you will need to use Trigonometry
formulas to solve hundreds of Mathematics problems. Also, these
formulas are only applicable to right -angled triangles.
With that being said, we will be mainly working with 3 sides of the right -
angled triangle: Perpendicular (Opposite side), Base (Adjacent side), and
Hypotenuse.
The Hypotenuse is the longest side, the Perpendicular is the opposite side
of the angle, and the base is where both Hypotenuse and Perpendicular
rest.
Now without making you wait any further, let us get on with our list of
Trigonometry formulas and their detailed explanation:
Basic Trigonometric Formulas
Before diving into the complex ones, le t’s go through the basic
trigonometry formulas first, also known as Trigonometric functions. There
are a total of six trigonometric functions that include sin, cos, tan, sec,
cot, and cosec.
3. Here are the formulas and trigonometric identities of these ratio s derived
by keeping a right-angled triangle as a reference:
• sin θ = Perpendicular (Opposite side) / Hypotenuse
• cos θ = Base (Adjacent side) / Hypotenuse
• tan θ = Perpendicular (Opposite side) / Base (Adjacent side)
• sec θ = Hypotenuse / Base (Adjacent side)
• cosec θ = Hypotenuse / Perpendicular (Opposite side)
• cot θ = Base (Adjacent side) / Perpendicular (Opposite side)
Reciprocal Identities
The reciprocal trigonometric identities are derived by keeping a right -
angled triangle as a reference and using the basic trigonometric ratios.
Here is a list of reciprocal identities and their formulas:
sin θ = 1/cosec θ
cos θ = 1/sec θ
tan θ = 1/cot θ
sec θ = 1/cos θ
cot θ = 1/tan θ
cosec θ = 1/sin θ
Trigonometric Ratio Table
We are compiling some of the most commonly used trigonometry formulas
and angles you will find in most trigonometric problems. Here it is:
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
4. 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
Periodic Identities
The trigonometry formulas for periodic identities are basically used to
shift the angles of π, π/2, 2π, etc. Moreover, these periodic and
trigonometry identities will keep repeating themselves are short intervals
of time. Why? Well, because they are known to be cyclic in nature.
Here are the trigonometry formulas of periodic identities according to
different quadrants, and in radians:
First Quadrant
• sin (π/2 – θ) = cos θ
• cos (π/2 – θ) = sin θ
• sin (π/2 + θ) = cos θ
• cos (π/2 + θ) = – sin θ
Second Quadrant
• sin (3π/2 – θ) = – cos θ
• cos (3π/2 – θ) = – sin θ
• sin (3π/2 + θ) = – cos θ
• cos (3π/2 + θ) = sin θ
5. Third Quadrant
• sin (π – θ) = sin θ
• cos (π – θ) = – cos θ
• sin (π + θ) = – sin θ
• cos (π + θ) = – cos θ
Fourth Quadrant
• sin (2π – θ) = – sin θ
• cos (2π – θ) = cos θ
• sin (2π + θ) = sin θ
• cos (2π + θ) = cos θ
Co-Function Identities (In Degrees)
The trigonometry formulas of co-function identities help you identify the
relationship between various trigonometry functions. Here is how we
present these trigonometry formulas in degrees:
• 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
Trigonometry Formulas for 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)
6. • 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)
Trigonometry Formulas for Double Angle Identities
• sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]
• cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(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
Trigonometry Formulas for Triple Angle Identities
• sin 3x = 3sin x – 4sin3x
• cos 3x = 4cos3x – 3cos x
• tan 3x = [3tanx – tan3x]/[1 – 3tan2x]
Trigonometry Formulas for Half Angle Identities
• sin (x/2) = ±√[(1 – cos x)/2]
• cos (x/2) = ± √[(1 + cos x)/2]
• tan (x/2) = ±√[(1 – cos x)/(1 + cos x)]
• or, tan (x/2) = ±√[(1 – cos x)(1 – cos x)/(1 + cos x)(1 – cos x)]
• tan (x/2) = ±√[(1 – cos x)2/(1 – cos2x)]
7. • ⇒ tan (x/2) = (1 – cos x)/sin x
Inverse Trigonometry Formulas
In these inverse trigonometry formulas, the value of x can be in fractions,
decimals, exponents, and whole numbers.
• 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
Major Systems of Trigonometry Formulas
Maybe you already got an idea about this, but let us explain you
thoroughly. All the trigonometric formulas are divided into two major
systems. Let us take a brief look on them:
Trigonometric Identities: Trigonometric identities are all the formulas that
are linked with Trigonometric functions.
Trigonometric Ratios: Trigonometric ratio is basically the relationship
between the side lengths of a right triangle and the measurement of the
angles.
Final Words
We tried to explain almost all trigonometric functions, different
trigonometric identities, trigonometry ratios, and almost all the formulas
8. in the post above. All of these above formulas will help you solve almost
all sorts of trigonometric problems.
Still, if you have any more questions or confusions, feel free to explore
our blog and we hope you will surely find your answers!
Frequently Asked Questions
What are the 6 basic trigonometric ratios?
The 6 basic trigonometric ratios are sin, cos, tan, cot, sec, and cosec.
What are the trigonometry formulas for Pythagorean identities?
• sin2A + cos2A = 1
• 1 + tan2A = sec2A
• 1 + cot2A = cosec2A
