Trigonometry Formulas for Class 12

Last Updated : 26 Dec, 2023

Trigonometry Formula for Class 12 is a compilation of all Trigonometry Formulas useful for Class 12 Students. This article contains all the formulas used in Trigonometry in one place that would help students appearing in Class 12 Board Exams as well as the JEE Exam for their last-minute revision and excel in the exams.

Trigonometry Formulas Class 12

Trigonometry Formulas Class 12 is a summary of all the formulas studied in class 12. These trigonometric formulas will help in solving all questions based on Trigonometry.

When learning about trigonometric formulas, we typically focus on right-angled triangles because these triangles provide a fundamental and simplified context for understanding and applying trigonometry. In a right-angled triangle, one of the angles is 90 degrees. There are 3 sides in a right-angled triangle – Hypotenuse (h), Perpendicular (P), and Base (B). The longest side is known as the Hypotenuse, the side opposite to the angle is Perpendicular and the side where both hypotenuse and opposite side intersect is the adjacent side (Base).

The list of the formulas of Trigonometry in class 12 is sequenced below in a serial order for ease of understanding.

• Basic Trigonometry Ratios
• Reciprocal Identities
• Trigonometry Identities
• Trigonometry Angles
• Co-function identities
• Compound Angles
• Sum to Product Identities
• Product to Sum Identities
• Half Angle Identities
• Double Angle Identities
• Triple Angle Identities
• Inverse Identities
• Properties of Inverse Trigonometric functions
• Addition Properties of Inverse trigonometric functions
• Twice and Thrice of Inverse Trigonometric functions

After this, for better understanding of yours, the list of all the above formulas are merged together in a single table.

Learn, Trigonometry Formulas

Basic Trigonometry Ratios

By visualizing the right-angled triangle image,

There are in total six trigonometry functions namely – sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), cosecant (cosec). The functions are:

• sin Î¸ = Perpendicular / Hypotenuse (P/H)
• cos Î¸ = Base / Hypotenuse (B/H)
• tan Î¸ = Perpendicular / Base (P/B)
• cot Î¸ = Base / Perpendicular (B/P)
• sec Î¸ = Hypotenuse / Base (H/B)
• cosec Î¸ = Hypotenuse / Perpendicular (H/P)

This formulas can be easily remembered by a trick which is mentioned below:

“Some People Have, Curly Brown Hairs, Turn Permanently Black”

Let’s take sin Î¸ = Perpendicular / Hypotenuse i.e. s for sin (Some), p for perpendicular (People), h for hypotenuse (Have). sin Î¸, cos Î¸ and tan Î¸ you can remember by this statement and cot Î¸ is the reciprocal of tan Î¸, sec Î¸ is the reciprocal of cos Î¸ and cosec Î¸ is the reciprocal of sin Î¸, which is mentioned in the below section.

Learn, Trigonometry Ratio

Reciprocal Identities

The reciprocal identities are mentioned below:

• sin Î¸ = 1 / cosec Î¸
• cos Î¸ = 1 / sec Î¸
• tan Î¸ = 1 / cot Î¸
• cot Î¸ = 1 / tan Î¸
• sec Î¸ = 1 / cos Î¸
• cosec Î¸ = 1 / sin Î¸

Trigonometry Identities

Below are some of the basic trigonometry identities:

• sin2 Î¸ + cos2 Î¸ = 1
• sec2 Î¸ – tan2 Î¸ = 1
• cosec2 Î¸ – cot2 Î¸ = 1
• tan Î¸ = sin Î¸ / cos Î¸
• cot Î¸ = cos Î¸ / sin Î¸
• tan Î¸ cot Î¸ = 1

Learn, Trigonometry Identities

Trigonometry Ratio Table

Below is the table for some basic trigonometry angles which will be used most frequently:

Angles (degree)

0Â°

30Â°

45Â°

60Â°

90Â°

180Â°

270Â°

360Â°

0

Ï€/6

Ï€/4

Ï€/3

Ï€/2

Ï€

3Ï€/2

2Ï€

sin

0

1/2

1/âˆš2

âˆš3/2

1

0

-1

0

cos

1

âˆš3/2

1/âˆš2

1/2

0

-1

0

1

tan

0

1/âˆš3

1

âˆš3

âˆž

0

âˆž

0

cot

âˆž

âˆš3

1

1/âˆš3

0

âˆž

0

âˆž

sec

1

2/âˆš3

âˆš2

2

âˆž

-1

âˆž

1

cosec

âˆž

2

âˆš2

2/âˆš3

1

âˆž

-1

âˆž

Co-function Identities

Below are some of the periodic identities

• sin(90Â°âˆ’Î¸) = cos Î¸
• cos(90Â°âˆ’Î¸) = sin Î¸
• tan(90Â°âˆ’Î¸) = cot Î¸
• cot(90Â°âˆ’Î¸) = tan Î¸
• sec(90Â°âˆ’Î¸) = cosec Î¸
• cosec(90Â°âˆ’Î¸) = sec Î¸

Compound Angles

Below are the formulas to express trigonometric functions as the sum or difference of two angles:

• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• sin(A â€“ B) = sin(A)cos(B) â€“ cos(A)sin(B)
• cos(A + B) = cos(A)cos(B) â€“ sin(A)sin(B)
• cos(A â€“ 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)

Sum to Product Identities

Below are some of the sum to product identities used:

• sin A + sin B = 2sin((A + B)/2) âˆ™ cos((A – B)/2)
• sin A – sin B = 2cos((A + B)/2) âˆ™ sin((A – B)/2)
• cos A + cos B = 2cos((A + B)/2) âˆ™ cos((A – B)/2)
• cos A – cos B = -2sin((A + B)/2) âˆ™ sin((A – B)/2)

Product to Sum Identities

Below are some of the product to sum identities used:

• sin A â‹… cos B = (sin(A + B) + sin(A – B)) / 2
• cos A â‹… sin B = (sin(A + B) – sin(A – B)) / 2
• sin A â‹… sin B = (cos(A – B) – cos(A + B)) / 2
• cos A â‹… cos B = (cos(A + B) + cos(A – B)) / 2

Half Angle Identities

Below are some half angle identities used:

• sin A/2 = Â± âˆš(1 – cos A) / 2
• cos A/2 = Â± âˆš(1 + cos A) / 2
• tan A/2 = Â± âˆš(1 – cos A) / (1 + cos A) = (1 – cos A) / sin A

Learn, Half Angle Identities

Double Angle Identities

Below are some double angle identities used:

• sin 2A = 2sin A cos A = 2tanA/(1+tan2A), {in terms of tan A}
• cos 2A = cos2A – sin2A = 2cos2A – 1 = 1 – 2sin2A = (1 – tan2A) / (1 + tan2A)
• tan 2A = 2tanA / (1 – tan2A)
• cot 2A = (cot2A – 1) / 2cotA
• sec 2A = sec2A / (2 – sec2A)
• cosec 2A = (sec A cosec A) / 2

Triple Angle Identities

Below are some triple angle identities used:

• sin 3A = 3sinA â€“ 4sin3A
• cos 3A = 4cos3A-3cos A
• tan 3A = (3tanA – tan3A) / (1 – 3tan2A)

Learn, Multiple Angle Formulas

Inverse Trig Formulas Class 12

Trigonometry in Class 12 mainly focuses on Inverse Trigonometry. Hence, we need to focus on all the formulas of Inverse Trigonometry. Following Inverse Trigonometry Formulas are discussed:

• Inverse Identities
• Properties of Inverse Trigonometric Functions
• Addition Properties of Inverse trigonometric functions
• Twice and Thrice of Inverse Trigonometric functions

Let’s learn these inverse trigonometry formulas for class 12 in detail below:

Inverse Identities

Below are some Inverse identities used:

• sin-1 (â€“A) = â€“ sin-1 A, A âˆˆ [-1,1]
• cos-1 (â€“A) = Ï€ â€“ cos-1 A, A âˆˆ [-1,1]
• tan-1 (â€“A) = â€“ tan-1 A, A âˆˆ R
• cot-1 (â€“A) = Ï€ â€“ cot-1 A, A âˆˆ R
• sec-1 (â€“A) = Ï€ â€“ sec-1 A, |A| â©¾ 1
• cosec-1 (â€“A) = â€“ cosec-1 A, |A| â©¾ 1

Properties of Inverse Trigonometric Functions

Below are some basic properties of inverse inverse trigonometric functions which will be used in solving many problems:

• sin-1(1/A) = cosec-1(A), if A â‰¥ 1 or A â‰¤ -1
• cos-1(1/A) = sec-1(A), if A â‰¥ 1 or A â‰¤ -1
• tan-1(1/A) = cot-1(A), A > 0
• sin(sin-1(A)) = A, âˆ’1â‰¤ A â‰¤1
• cos(cos-1(A)) = A, âˆ’1â‰¤ A â‰¤1
• tan(tan-1(A)) = A,â€“âˆž< A <âˆž
• cot(cot-1(A)) = A, â€“âˆž< A <âˆž
• sec(sec-1(A)) = A, âˆ’âˆž< A â‰¤1 or 1â‰¤ A <âˆž
• cosec(cosec-1(A)) = A, â€“âˆž< A â‰¤1 or âˆ’1â‰¤ A <âˆž

Addition Properties of Inverse Trigonometric Functions

Below are some the addition properties of inverse inverse trigonometric functions:

• sin-1A + cos-1A = Ï€/2 , A âˆˆ [-1, 1]
• tan-1A + cot-1A = Ï€/2 , A âˆˆ R
• sec-1A + cosec-1A = Ï€/2 ,|A| â‰¥ 1
• tan-1 A + tan-1 B = tan-1((A + B)/(1 – AB)), if the value AB < 1
• tan-1 A â€“ tan-1 B = tan-1((A – B)/(1 + AB)), if the value AB > -1

Twice and Thrice of Inverse Trigonometric Functions

Below are some of the twice and thrice inverse trigonometric properties:

• 2tan-1 A = sin-1(2A / (1 + A2)), |A| â‰¤ 1
• 2tan-1 A = cos-1((1 – A2) / (1 + A2)), A â‰¥ 0
• 2tan-1 A = tan-1(2A / (1 – A2)), -1< A <1
• 3sin-1 A = sin-1(3A – 4A3)
• 3cos-1 A = cos-1(4A3 – 3A)
• 3tan-1 A = tan-1((3A – A3)/(1 – 3A2))

Table of Trigonometry Formulas Class 12

The following table contains the formulas of Trigonometry Class 12:

Trigonometry Formulas Class 12 Table

Name of Identity

Formulas

Inverse Identities

sin-1 (â€“A) = â€“ sin-1 A, A âˆˆ [-1,1]

cos-1 (â€“A) = Ï€ â€“ cos-1 A, A âˆˆ [-1,1]

tan-1 (â€“A) = â€“ tan-1 A, A âˆˆ R

cot-1 (â€“A) = Ï€ â€“ cot-1 A, A âˆˆ R

sec-1 (â€“A) = Ï€ â€“ sec-1 A, |A| â©¾ 1

cosec-1 (â€“A) = â€“ cosec-1 A, |A| â©¾ 1

Properties of Inverse Trigonometric functions

sin-1(1/A) = cosec-1(A), if A â‰¥ 1 or A â‰¤ -1

cos-1(1/A) = sec-1(A), if A â‰¥ 1 or A â‰¤ -1

tan-1(1/A) = cot-1(A), A > 0

sin(sin-1(A)) = A, âˆ’1â‰¤ A â‰¤1

cos(cos-1(A)) = A, âˆ’1â‰¤ A â‰¤1

tan(tan-1(A)) = A,â€“âˆž< A <âˆž

cot(cot-1(A)) = A, â€“âˆž< A <âˆž

sec(sec-1(A)) = A, âˆ’âˆž< A â‰¤1 or 1â‰¤ A <âˆž

cosec(cosec-1(A)) = A, â€“âˆž< A â‰¤1 or âˆ’1â‰¤ A <âˆž

Addition Properties of Inverse Trigonometric functions

sin-1A + cos-1A = Ï€/2 , A âˆˆ [-1, 1]

tan-1A + cot-1A = Ï€/2 , A âˆˆ R

sec-1A + cosec-1A = Ï€/2 ,|A| â‰¥ 1

tan-1 A + tan-1 B = tan-1((A + B)/(1 – AB)), if the value AB < 1

tan-1 A â€“ tan-1 B = tan-1((A – B)/(1 + AB)), if the value AB > -1

Twice and Thrice of Inverse Trigonometric functions

2tan-1 A = sin-1(2A / (1 + A2)), |A| â‰¤ 1

2tan-1 A = cos-1((1 – A2) / (1 + A2)), A â‰¥ 0

2tan-1 A = tan-1(2A / (1 – A2)), -1< A <1

3sin-1 A = sin-1(3A – 4A3)

3cos-1 A = cos-1(4A3 – 3A)

3tan-1 A = tan-1((3A – A3)/(1 – 3A2))

Also, Check

Trigonometry Formula Class 12 – Solved Examples

Example 1. Find the value of cos-1(0)?

Solution:

Let cos-1(0) = x

â‡’ cos x = 0

â‡’ cos x = 90

Hence, x = 90

Thus, cos-1(0) = 90Â°

Example 2. Find the value of sin Ï€/2 + cos Ï€/2?

Solution:

Since, sin Ï€/2 = 1

Cos Ï€/2 = 0

â‡’ sin Ï€/2 + cos Ï€/2 = 1 + 0 = 1

Example 3. Simplify tan x cot x.

Solution:

Since, tan x = 1/cot x

â‡’ tan x. cot x = tan x.1/tan x = 1

Example 4. Given sec x = 5/3. Find cos x.

Solution:

cos x = 1 / sec x

â‡’ cos x = 1 / (5/3)

â‡’ cos x = 3/5

Example 5. If sin x = -4/5, find the value of cosec x.

Solution:

cosec x = 1 / sin x

â‡’ cosec x = 1 / (-4/5)

â‡’ cosec x = -5/4

Example 6. Find sin(120Â°)

Solution:

Sin 120 = sin(2 Ã— 60)

As per sin2Î¸ = 2 sinÎ¸ cosÎ¸

Here, Î¸ = 60Â°

â‡’ Sin 120 = 2sin60Â° cos 60Â°

â‡’ Sin 120 = 2Ã—âˆš3/2Ã—1/2 = âˆš3/2

Trigonometry Formulas Class 12 – Practice Questions

Try out following practice questions based on Trigonometry Formula Class 12

Q1. Prove that (sin4Î¸ â€“ cos4Î¸ +1) cosec2Î¸ = 2.

Q2. Evaluate: 2 tan245Â° + cos230Â° â€“ sin260Â°.

Q3. Find the principal values of Sin-1(1/âˆš2).

Q4. Find the value of cos(sin-1 x + cos-1 x).

Q5. Find the value of sin-1 (sin 4Ï€/5 ).

Trigonometry Formula Class 12 – FAQs

1. How many Trigonometric Functions are there?

There are six Trigonometric Functions namely sine, cosine, secant, cosecant, tangent and cotangent.

2. What are Inverse Trigonometric Functions?

For a Trigonometric Function, let say sin Î¸ = x, inverse trigonometric function is given as Î¸ = sin-1x

3. How many Inverse Trigonometric Functions are there?

There are six inverse trigonometric functions namely arcsine, arccos, arcsec, arccosec, arctan, arccot

4. What is Trigonometry?

Trigonometry is the Branch of Mathematics that establish the relation between the sides and angles of a triangle

5. Who is the Father of Trigonometry?

Hipparchus of Nicaea was a Greek Mathematician, Astronomer, Geographer from 190 BC. He is considered the founder of trigonometry.

6. How to Memorize Trigonometry Formula Class 12?

To memorizre Trigonometry Formula Class 12 read the formulas mentioned in this article section wise and solve questions based on them.

7. What is Trigonometry Formulas PDF?

The trigonometry formula pdf is a pdf file that contains all the formulas of trigonometry that can be downlowded from our website.

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