✅ Trigonometry Formula ⭐️⭐️⭐️⭐️⭐

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Basic Trigonometric Function Formulas

There are basically 6 ratios used for finding the elements in Trigonometry. They are called trigonometric functions. The six trigonometric functions are sine, cosine, secant, co-secant, tangent and co-tangent.

By using a right-angled triangle as a reference, the trigonometric functions and identities are derived:

  • 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

The Reciprocal Identities are given 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

All these are taken from a right angled triangle. When height and base side of the right triangle are known, we can find out the sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas. The reciprocal trigonometric identities are also derived by using the trigonometric functions.

Cosines and sines around the unit circle

Trigonometry Table

Below is the table for trigonometry formulas for angles that are commonly used for solving problems.

Angles (In Degrees)30°45°60°90°180°270°360°
Angles (In Radians)π/6π/4π/3π/2π3π/2
sin01/21/√2√3/210-10
cos1√3/21/√21/20-101
tan01/√31√300
cot√311/√300
csc2√22/√31-1
sec12/√3√22-11

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

All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities. tan 45° = tan 225° but this is true for cos 45° and cos 225°. Refer to the above trigonometry table to verify the values.

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

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)

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) = secx/(2-sec2 x)
  • csc (2x) = (sec x. csc x)/2

Triple Angle Identities

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

Half Angle Identities

Product identities

Sum to Product Identities

Inverse Trigonometry Formulas

What is Sin 3x Formula?

Sin 3x is the sine of three times of an angle in a right-angled triangle, that is expressed as:

Sin 3x = 3sin x – 4sin3x

Trigonometry Formulas From Class 10 to Class 12

Trigonometry Formulas For Class 10

The trigonometric formulas for ratios are majorly based on the three sides of a right-angled triangle, such as the adjacent side or base, perpendicular and hypotenuse (See the above figure).  Applying Pythagoras theorem for the given right-angled triangle, we have:

(Perpendicular)2+(Base)2=(Hypotenuse)2

⇒(P)2+(B)2=(H)2

Now, let us see the formulas based on trigonometric ratios (sine, cosine, tangent, secant, cosecant and cotangent)

Basic Trigonometric formulas

The Trigonometric formulas are given below:

S.noPropertyMathematical value
1sin APerpendicular/Hypotenuse
2cos ABase/Hypotenuse
3tan APerpendicular/Base
4cot ABase/Perpendicular
5cosec AHypotenuse/Perpendicular
6sec AHypotenuse/Base

Reciprocal Relation Between Trigonometric Ratios

S.noIdentityRelation
1tan Asin A/cos A
2cot Acos A/sin A
3cosec A1/sin A
4sec A1/cos A

Trigonometric Sign Functions

  • sin (-θ) = − sin θ
  • cos (−θ) = cos θ
  • tan (−θ) = − tan θ
  • cosec (−θ) = − cosec θ
  • sec (−θ) = sec θ
  • cot (−θ) = − cot θ

Trigonometric Identities

  1. sin2A + cos2A = 1
  2. tan2A + 1 = sec2A
  3. cot2A + 1 = cosec2A

Periodic Identities

  • sin(2nπ + θ ) = sin θ
  • cos(2nπ + θ ) = cos θ
  • tan(2nπ + θ ) = tan θ
  • cot(2nπ + θ ) = cot θ
  • sec(2nπ + θ ) = sec θ
  • cosec(2nπ + θ ) = cosec θ

Complementary Ratios

Quadrant I

  • sin(π/2−θ) = cos θ
  • cos(π/2−θ) = sin θ
  • tan(π/2−θ) = cot θ
  • cot(π/2−θ) = tan θ
  • sec(π/2−θ) = cosec θ
  • cosec(π/2−θ) = sec θ

Quadrant II

sin(π−θ) = sin θ
cos(π−θ) = -cos θ
tan(π−θ) = -tan θ
cot(π−θ) = – cot θ
sec(π−θ) = -sec θ
cosec(π−θ) = cosec θ

Quadrant III

  • sin(π+ θ) = – sin θ
  • cos(π+ θ) = – cos θ
  • tan(π+ θ) = tan θ
  • cot(π+ θ) = cot θ
  • sec(π+ θ) = -sec θ
  • cosec(π+ θ) = -cosec θ

Quadrant IV

  • sin(2π− θ) = – sin θ
  • cos(2π− θ) = cos θ
  • tan(2π− θ) = – tan θ
  • cot(2π− θ) = – cot θ
  • sec(2π− θ) = sec θ
  • cosec(2π− θ) = -cosec θ

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) = [(tan A + tan B)/(1 – tan A tan B)]
  • tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)]

Double Angle Formulas

  • sin2A = 2sinA cosA = [2tan A + (1+tan2A)]
  • cos2A = cos2A–sin2A = 1–2sin2A = 2cos2A–1= [(1-tan2A)/(1+tan2A)]
  • tan 2A = (2 tan A)/(1-tan2A)

Triple Angle Formulas

  • sin3A = 3sinA – 4sin3A
  • cos3A = 4cos3A – 3cosA
  • tan3A = [3tanA–tan3A]/[1−3tan2A]

Trigonometry Formulas For Class 11

List of Class 11 Trigonometry Formulas

Here is the list of formulas for Class 11 students as per the NCERT curriculum. All the formulas of trigonometry chapter are provided here for students to help them solve problems quickly.

Trigonometry Formulas
sin(−θ) = −sin θ
cos(−θ) = cos θ
tan(−θ) = −tan θ
cosec(−θ) = −cosecθ
sec(−θ) = sec θ
cot(−θ) = −cot θ
Product to Sum Formulas
sin x sin y = 1/2 [cos(x–y) − cos(x+y)]
cos x cos y = 1/2[cos(x–y) + cos(x+y)]
sin x cos y = 1/2[sin(x+y) + sin(x−y)]
cos x sin y = 1/2[sin(x+y) – sin(x−y)]
Sum to Product Formulas
sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2]
sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2]
cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2]
cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2]
Identities
sin2 A + cos2 A = 1
1+tan2 A = sec2 A
1+cot2 A = cosec2 A

Sign of Trigonometric Functions in Different Quadrants

Quadrants→IIIIIIIV
Sin A++
Cos A++
Tan A++
Cot A ++
Sec A++
Cosec A ++

Basic Trigonometric Formulas for Class 11

cos (A + B) = cos A cos B – sin 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

sin (A -B) = sin A cos B – cos A sin B

Based on the above addition formulas for sin and cos, we get the following below formulas:

  • sin(π/2-A) = cos A
  • cos(π/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

If none of the angles A, B and (A ± B) is an odd multiple of π/2, then

  • tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)]
  • tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)]

If none of the angles A, B and (A ± B) is a multiple of π, then

  • cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)]
  • cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)]

Some additional formulas for sum and product of angles:

  • cos(A+B) cos(A–B)=cos2A–sin2B=cos2B–sin2A
  • sin(A+B) sin(A–B) = sin2A–sin2B=cos2B–cos2A
  • sinA+sinB = 2 sin (A+B)/2 cos (A-B)/2

Formulas for twice of the angles:

  • sin2A = 2sinA cosA = [2tan A /(1+tan2A)]
  • cos2A = cos2A–sin2A = 1–2sin2A = 2cos2A–1= [(1-tan2A)/(1+tan2A)]
  • tan 2A = (2 tan A)/(1-tan2A)

Formulas for thrice of the angles:

  • sin3A = 3sinA – 4sin3A
  • cos3A = 4cos3A – 3cosA
  • tan3A = [3tanA–tan3A]/[1−3tan2A]

Trigonometry Formulas For Class 12

Basic Trigonometric Functions

  • Sin θ = Opposite side of angle θ/Hypotenuse
  • Cos θ = Adjacent side of angle θ/Hypotenuse
  • Tan θ = Opposite side of angle θ/Adjacent side of angle θ
  • Sec θ = Hypotenuse/Adjacent side of angle θ
  • Cosec θ = Hypotenuse/Opposite side of angle θ
  • Cot θ = Adjacent side of angle θ/Opposite side of angle θ

Domain and Range of Trigonometric Functions

Here are the domain and range of basic trigonometric functions:

  • Sine function, sine: R → [– 1, 1]
  • Cosine function, cos : R → [– 1, 1]
  • Tangent function, tan : R – { x : x = (2n + 1) π/2, n ∈ Z} →R
  • Cotangent function, cot : R – { x : x = nπ, n ∈ Z} →R
  • Secant function, sec : R – { x : x = (2n + 1) π/2, n ∈ Z} →R – (– 1, 1)
  • Cosecant function, cosec : R – { x : x = nπ, n ∈ Z} →R – (– 1, 1)

Properties of Inverse Trigonometric Functions

  • sin-1 (1/a) = cosec-1(a), a ≥ 1 or a ≤ – 1
  • cos-1(1/a) = sec-1(a), a ≥ 1 or a ≤ – 1
  • tan-1(1/a) = cot-1(a), a>0
  • sin-1(–a) = – sin-1(a), a ∈ [– 1, 1]
  • tan-1(–a) = – tan-1(a), a ∈ R
  • cosec-1(–a) = –cosec-1(a), | a | ≥ 1
  • cos-1(–a) = π – cos-1(a), a ∈ [– 1, 1]
  • sec-1(–a) = π – sec-1(a), | a | ≥ 1
  • cot-1(–a) = π – cot-1(a), a ∈ R

Addition Properties of Inverse Trigonometry functions

  • sin-1a + cos-1a = π/2, a ∈ [– 1, 1]
  • tan-1a + cot-1a = π/2, a ∈ R
  • cosec-1a + sec-1a = π/2, | a | ≥ 1
  • tan-1a + tan-1 b = tan-1 [(a+b)/1-ab], ab<1
  • tan-1a – tan-1 b = tan-1 [(a-b)/1+ab], ab>-1
  • tan-1a – tan-1 b = π + tan-1[(a+b)/1-ab], ab > 1; a,b > 0

Twice of Inverse of Tan Function

  • 2tan-1a = sin-1 [2a/(1+a2)], |a| ≤ 1
  • 2tan-1a = cos-1[(1-a2)/(1+a2)], a ≥ 0
  • 2tan-1a = tan-1[2a/(1+a2)], – 1 < a < 1

These are important formulas introduced in the Inverse trigonometric functions chapter of Class 12. Students can solve the problems based on these properties taking reference from this article.

Sine and Cosine Laws

Sine laws: The sine law and the cosine law give us the relationship between the sides and the angles of a triangle. The Sine Law gives the ratio of the sides and the angle opposite to the side. 

Example: The ratio can be taken for the side a and its opposite angle ‘A’.

Cosine laws: The cosine Law helps to find the length of a side for the given lengths of the other two sides and the included angle.  For example, the length ‘a′‘a′ can be found with the help of sides bb and c,c, and their included angle A.A.
a2=b2+c2–2bccosA
b2=a2+c2–2accosB
c2=a2+b2–2abcosC
a,b and c are the lengths of sides of the triangle, and A,B,C are the angles of the triangle.

Trigonometry Formulas Major systems

All trigonometric formulas are divided into two major systems:

  • Trigonometric Identities
  • Trigonometric Ratios

Trigonometric Identities are formulas that involve Trigonometric functions. These identities are true for all values of the variables. Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the sides of the right triangle.

Here we provide a list of all Trigonometry formulas for the students. These formulas are helpful for the students in solving problems based on these formulas or any trigonometric application. Along with these, trigonometric identities help us to derive the trigonometric formulas, if they will appear in the examination.

We also provided the basic trigonometric table pdf that gives the relation of all trigonometric functions along with their standard values. These trigonometric formulae are helpful in determining the domain, range, and value of a compound trigonometric function. Students can refer to the formulas provided below or can also download the trigonometric formulas pdf that is provided above.

Solved Problems

Q.1:What is the value of (sin30° + cos30°) – (sin 60° + cos60°)?

Sol: Given,

(sin30° + cos30°) – (sin 60° + cos60°)

= ½ + √3/2 – √3/2 – ½

= 0

Q.2: If cos A = 4/5, then tan A = ?

Sol: Given, 

Cos A = ⅘

As we know, from trigonometry identities,

1+tan2A = sec2A

sec2A – 1 = tan2A

(1/cos2A) -1 = tan2A

Putting the value of cos A = ⅘.

(5/4)2 – 1 = tan2 A

tan2A = 9/16

tan A = 3/4

Examples Using Trigonometry Formulas

Example 1: Rachel is given the trigonometric ratio of tan θ = 5/12. Help Rachel to find the trigonometric ratio of cosec θ using trigonometry formulas.

Solution:

tan θ = Perpendicular/ Base = 5/12

Perpendicular = 5 and Base = 12

Hypotenuse2 = Perpendicular2 + Base2

Hypotenuse2 = 52 + 122

Hypotenuse2 = 25 + 144

Hypotenuse = √169

Hypotenuse = 13

Hence, sin θ = Perpendicular/Hypotenuse = 5/13

cosec θ = Hypotenuse/Perpendicular = 13/5

Answer: Using trigonometry formulas, cosec θ = 13/5

Example 2: As part of the assignment, Samuel has to find the value of Sin 15º using the trigonometry formulas. How can we help Samuel to find the value?
Solution:

sin 15º

= sin (45º – 30º)

= sin 45ºcos 30º – cos 45ºsin 30º

= [(1/√2) × (√3/2)] – [(1√2) × (1/2)] = (√3 – 1)/2√2

Answer: sin 15° = (√3 – 1)/2√2

Example 3: If sin θcos θ = 5, find the value of (sin θ + cos θ)using the trigonometry formulas.

Solution:

(sin θ + cos θ)2

= sin2θ + cos2θ + 2sinθcosθ

= (1) + 2(5) = 1 + 10 = 11

Answer: (sin θ + cos θ)= 11

Frequently Asked Questions – FAQs

What are the basic trigonometric ratios?

Sine, Cosine, Tangent, Cotangent, Secant and Cosecant.

What are formulas for trigonometry ratios?

Sin A = Perpendicular/Hypotenuse
Cos A = Base/Hypotenuse
Tan A = Perpendicular/Base

What are the three main functions in trigonometry?

Sin, Cos and Tan are three main functions in trigonometry.

What are the fundamental trigonometry identities?

The three fundamental identities are:
1. sin2 A + cos2 A = 1
2. 1+tan2 A = sec2 A
3. 1+cot2 A = csc2 A

Trigonometry formulas are applicable to which triangle?

Right-angled triangle

Q.1. What is the basic formula of trigonometry?
Ans: 
We have six fundamental trigonometric ratios used in Trigonometry. These ratios are also known as trigonometric functions. The six essential trigonometric functions are sine, cosine, secant, cosecant, tangent, and cotangent. The trigonometric functions and identities are derived by using the right-angled triangle. When the height and the base side of the right triangle are known, we can find the sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas.

Q.2. What are the three formulas of trigonometry?
Ans:
 
The three formulas of trigonometry are sine, cosine and tangent.
The formulas are given below:

Q.3. What are the formulas for trigonometry ratios?
Ans: 
The trigonometric ratios formulas are written below:

Q.4. What are the 11 trigonometric identities?
Ans:
 The eleven trigonometric identities in trigonometry are given below:
1. Basic formulas
2. Reciprocal Identities
3. Trigonometric Ratio Table
4. Periodic Identities
5. Co-function Identities
6. Sum and Difference Identities
7. Half-Angle Identities
8. Double Angle Identities
9. Triple Angle Identities
10. Product Identities
11. Sum of Product Identities

Q.5. How to remember trigonometry formulas class 11?
Ans:
 We have many formulas in the higher classes that might be difficult to remember, so there are few steps to follow for remembering them:
1. Get familiar with mathematical symbols.
2. Then comes the structure of the formulas and how are they derived.
3. Practice the formulas regularly. 
4. Use flashcards of the formulas, then revise and finally test yourself.

Q.6. What are three Pythagorean identities?
Ans:
 The first identity states that sine squared plus cosine squared identical one. The second one states that tangent squared plus one is similar to secant squared. For the last one, it says that one plus cotangent squared is comparable to cosecant squared.
Formulas are:
θ+θ=1
θ+1=θ
1+θ=θ

What are Trigonometry Formulas?

Trigonometry formulas are used to solve problems based on the sides and angles of a right-angled triangle, using the different trigonometric identities. These formulas can be used to evaluate trigonometric ratios(also referred to as trigonometric functions), sin, cos, tan, csc, sec, and cot.

What is the Basic Trigonometry Formula?

Basic trigonometry formulas involve the representing of basic trigonometric ratios in terms of the ratio of corresponding sides of a right-angled triangle. These are given as, sin θ = Opposite Side/Hypotenuse, cos θ = Adjacent Side/Hypotenuse, tan θ = Opposite Side/Adjacent Side.

What are Trigonometry Ratios’ Formulas?

The three main functions in trigonometry are Sine, Cosine, and Tangent. Trigonometry ratios’ formulas are given as,

  • Sine Function: sin(θ) = Opposite / Hypotenuse
  • Cosine Function: cos(θ) = Adjacent / Hypotenuse
  • Tangent Function: tan(θ) = Opposite / Adjacent

What are Trigonometry Formulas for Even and Odd Identities?

The trigonometry formulas involving even and odd identities are given as,

  • sin(–x) = –sin x
  • cos(–x) = cos x
  • tan(–x) = –tan x
  • csc (–x) = –csc x
  • sec (–x) = sec x
  • cot (–x) = –cot x

What are the Trigonometry Formulas Involving Pythagorean Identities?

The three fundamental trigonometry formulas involving the Pythagorean identities are given as,

  • sin2A + cos2A = 1
  • 1 + tan2A = sec2A
  • 1 + cot2A = cosec2A

Trigonometry Formulas Are Applicable to Which Triangle?

Trigonometry formulas are applicable to right-angled triangles. These trig formulas represent the trigonometric ratios in terms of the ratio of corresponding sides of a right-angled triangle.

What are Addition Trigonometry Formulas?

The trigonometry formulas for trigonometry ratios when the angles are in addition are given as,

  • 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)

How to Remember Trigonometry Formulas Easily?

The trick to learn basic trigonometry formulas is using the mnemonic “SOHCAHTOA”, which can be used to memorize trigonometric ratios as,
SOH: Sine = Opposite / Hypotenuse
CAH: Cosine = Adjacent / Hypotenuse
TOA: Tangent = Opposite / Adjacent

What is sin 3x Trigonometry Formula?

Trigonometry formula, sin 3x is the sine of three times of an angle in a right-angled triangle, it is expressed as: sin 3x = 3sin x – 4sin3x.

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