Trigonometry Formulas For Class 12

Trigonometry is the study between the relationships dealing with angles, heights and lengths of triangles and also the relationships between the different circle parts and other geometric figures. In trigonometry class 12, we study trigonometry which finds its application in the field of astronomy, engineering, architectural design, and physics.Trigonometry Formulas for class 12 contains all the essential trigonometric identities which can fetch some direct questions in competitive exams on the basis of formulae.


Trigonometric identities given in the 12th trigonometry formula are very useful and help to solve the problems better. There are huge numbers of fields in which these trigonometry identities and trigonometric equations are used.


The Difference Between Trigonometric Identities And Trigonometric Ratios:

Trigonometric Identities: Formula of trigonometry class 12 involves trigonometric functions and trigonometric identities. These identities of trigonometry are accurate for all variable’s values.

Trigonometric Ratio: The relationship of the angle measurement and the right-angle triangle side length is known for its trigonometric ratio.


All Trigonometry Formulas For Class 12

Sum And Difference Of Angles:

  1. Sin A Sin B = ½  [cos(A-B)-cos(A+B)]

  2. cos A cos B = ½ [cos(A-B)+ cos(A+B)]

  3. Sin A cos B = ½ [sin(A+B)+ sin(A-B)]

  4. cos A sin B = ½ [cos(A+B)+ sin(A-B)]


Double Angle Formulas:

  1. sin2θ= 2sinθcosθ

  2. cos2θ= Cos²θ - Sin²θ

                       = 2Cos²θ - 1

                       = 1-2Sin²θ

  1. tan2θ= \[\frac{2Tanθ}{1-Tan^{2}θ}\]


Triple Angle Formula:

  1. sin3θ= 3Sinθ - 4\[Sin^{3}θ\]

  2. cos3θ= 4\[Cos^{3}θ\] - 3Cosθ

  3. tan3θ= (3tanθ-tan³θ) / (1-3tan²θ)

  4. cot3θ= \[\frac{Cot^{3}θ - 3Cotθ}{3Cot^{2}θ - 1}\]


Converting Products Into Sums And Difference:

  1. sinA sinB = ½ [cos(A-B) - cos(A+B)]

  2. cosA cosB = ½ [cos(A-B)+cos(A+B)]

  3. sinA cosB = ½ [sin(A+B)+sin(A-B)]

  4. cosA sinB = ½ [cos(A+B)+sin(A-B)]


Half Angle Identities:

  1. Sin(\[\frac{x}{2}\]) = ±\[\sqrt{\frac{1-2cos(x)}{2}}\]

  2. Tan(\[\frac{x}{2}\]) = ±\[\sqrt{\frac{1-cos(x)}{1+Cos(x)}}\] = \[\frac{1-Cos(x)}{Sin(x)}\] = \[\frac{Sin(x)}{1+Cos(x)}\]

  3. Cos(\[\frac{x}{2}\]) = ±\[\sqrt{\frac{1+cos(x)}{2}}\]


On squaring the above identities we can re-state the above equations as following-

  1. \[Sin^{2}(x)\] = \[\frac{1}{2}\][1-Cos(2x)]

  2. \[Cos^{2}(x)\] = \[\frac{1}{2}\][1+Cos(2x)]

  3. \[Tan^{2}(x)\] = \[\frac{1-Cos(2x)}{1+Cos(2x)}\]

Complex Relations:

  1. Sinθ = \[\frac{e^{iθ} - e^{-iθ}}{2i}\]

  2. Cosθ = \[\frac{e^{iθ} + e^{-iθ}}{2i}\]

  3. Tanθ = \[\frac{e^{iθ} - e^{-iθ}}{(e^{iθ} + e^{-iθ})i}\]

  4. Cosecθ = \[\frac{2i}{e^{iθ} - e^{-iθ}}\]

  5. Secθ =\[\frac{2i}{e^{iθ} - e^{-iθ}}\] 

  6. Cotθ =\[\frac{(e^{iθ} + e^{-iθ})i}{e^{iθ} - e^{-iθ}}\]


Inverse Trigonometric Functions:

Definition:

θ = \[Sin^{−1}(x)\] is equivalent to x = Sin θ

θ = \[Cos^{−1}(x)\] is equivalent to x = Cos θ

θ = \[Tan^{−1}(x)\] is equivalent to x = Tan θ


Inverse Properties:

These properties hold for x in the domain and θ in the range

sin(sin−1 (x)) = x 

cos(cos−1 (x)) = x

tan(tan−1 (x)) = x

sin−1 (sin(θ)) = θ 

cos−1 (cos(θ)) = θ

tan−1 (tan(θ)) = θ


Other Notations

sin−1 (x) = arcsin(x) 

cos−1 (x) = arccos(x)

tan−1 (x) = arctan(x)


Domain and Range:

Function

Domain

Range

θ = sin−1 (x)

−1 ≤ x ≤ 1

− π /2 ≤ θ ≤ π/2

θ = cos−1 (x)

−1 ≤ x ≤ 1

0 ≤ θ ≤ π

θ = tan−1 (x)

−∞ ≤ x ≤ ∞

− π/ 2 < θ < π/2


Inverse Trigonometric Functions:

Name

Usual Notation

Definition

Domain of x for real number

Range of usual principal values

(Radians)

Range of usual principal values

(Degrees)

arcsine

y=arcsin(x)

x=sin(y)

−1 ≤ x ≤ 1

− π /2 ≤ y ≤ π/2

−90⁰ ≤ y ≤ 90⁰

arccosine

y=arcos(x)

x=cos(y)

−1 ≤ x ≤ 1

0 ≤ y ≤ π

0⁰ ≤ y ≤ 180⁰

arctangent

y=arctan(x)

x=tan(y)

All real numbers

− π /2 ≤ y ≤ π/2

−90⁰ ≤ y ≤ 90⁰

arccotangent

y=arccot(x)

x=cot(y)

All real numbers

0 ≤ y ≤ π

0⁰ ≤ y ≤ 180⁰

arcsecant

y=arcsec(x)

x=sec(y)

x ≤ -1 or 1 ≤ x

0 ≤ y < π/2 or π/2 < y ≤π

0 ≤ y < 90⁰ or 90⁰ < y ≤180⁰

arccosecant

y=arccosec(x)

x=cosec(x)

x ≤ -1 or 1 ≤ x

- π/2 ≤ y < 0 or 0 < y ≤ π/2

- 90⁰ ≤ y < 0 or 0 < y ≤ 90⁰


Solved Examples:

Question 1) Find the principal values of \[Sin^{-1}\] (\[\frac{1}{\sqrt{2}}\])

Solution) Let \[Sin^{-1}\](\[\frac{1}{\sqrt{2}}\]) = α; then sin α =\[\frac{1}{\sqrt{2}}\]= sin 45 degree

α = 45 degree or \[\frac{π}{4}\], which is the required principal value.


Question 2) Show that \[Tan^{-1}\]\[\frac{1}{2}\] + \[Tan^{-1}\]\[\frac{1}{3}\] = \[\frac{π}{4}\] 

Solution) L.H.S. = \[Tan^{-1}\] \[\frac{1}{2}\] + \[Tan^{-1}\] \[\frac{1}{3}\] = \[Tan^{-1}\] = \[\frac{\frac{1}{2} + \frac{1}{3}}{1- \frac{1}{2} X \frac{1}{3}}\] = \[Tan^{-1}\] (\[\frac{\frac{5}{6}}{\frac{5}{6}}\]) = \[Tan^{-1}\] = \[\frac{π}{4}\]


Fun Facts

  • The word "Trigonometry" is taken from the word "Triangle Measure".

  • Trigonometry is used by the engineers to figure out the angles of the sound waves and how to design rooms.

  • Trigonometry is connected with music and architecture.

FAQ (Frequently Asked Questions)

1. What does quotient relations mean?

In trigonometry class 12, we come across quotient relations. Quotient relations comprise three trigonometric ratios. One is the quotient that we get after division operation between the other two. For example, tan θ = sin θ /cos θ and cot θ = cos θ / sin θ.

2. Who is the father of trigonometry?

Trigonometry is the study between the relationships dealing with angles, heights and lengths of triangles and also the relationships between the different circle parts and other geometric figures. Hipparchus, a Greek astronomer, is considered as the father of trigonometry.  12th Trigonometry formulas contain some trigonometry and its tables were developed by him. Hipparchus was the first person whose quantitative and accurate models for the motion of the sun and moon survive. Also, the first accurate star map is formulated by him. He may have been the first to develop a reliable method to predict a solar eclipse with the help of his solar and lunar theories.

3. What are the laws of trigonometry?

Trigonometry class 12 deals with trigonometric laws of sines and cosines which is helpful in solving a triangle i.e., finding all three angles and sides of any triangle  given three out of the six. The law of sines states that in any triangle, not just a right angle triangle but the ratio of the sine of each angle to its opposing side is also equivalent for all three angles. The law of cosines states that in any triangle, the sum of the squares of the other two sides is equal to the square of one side. The trigonometric law of cosines is helpful in solving a triangle if all three sides, or two sides and the included angle are given.