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Trigonometry Formulas For Class 12

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Last updated date: 28th Mar 2024
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Introduction

Trigonometry Formulas For Class 12 provided at Vedantu are sets of different formulas involving trigonometric identities, used to solve maths problems based on the sides and angles of a right-angled triangle. These Trigonometry Formulas For Class 12include trigonometric functions like sine, cosine, tangent, cosecant, secant, cotangent for given angles. These Trigonometry Formulas are very useful for students from class 12 which cover Pythagorean identities, product identities, cofunction identities (shifting angles), sum & difference identities, double angle identities, half-angle identities, etc. in detail to score good marks in Maths Class 12 board exam.

 

What is Trigonometry?

Trigonometry is the study of 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 based on 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: Equalities in trigonometry functions are known as trigonometric identities.  

  • 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

Double Angle Formulas:

  1. \[sin2\theta = 2sin\theta cos\theta \]

  2. \[cos2\theta = Cos^2 \theta - Sin^2 \theta = 2Cos^2 \theta - 1= 1 - 2Sin^2 \theta\]

  3. \[tan2\theta = \frac{ 2tan\theta }{1 - tan^2\theta} \]

 

Triple Angle Formula:

  1. \[sin3\theta = 3Sin\theta - 4 Sin^3 \theta\]

  2. \[cos\theta = 4 Cos^3 \theta  – 3Cos\theta \]

  3. \[tan3\theta = \frac {3tan \theta - tan^3 \theta }{1 - 3tan^2 \theta} \]

  4. \[cot3\theta = \frac {cot^3\theta - 3cot \theta }{3 cot^2 \theta - 1} \] 

 

Converting Products Into Sums And Difference:

  1. \[sinA sinB = \frac{1}{2} \begin{bmatrix} cos(A-B) -  cos(A+B) \end{bmatrix}\]

  2. \[cosA cosB = \frac{1}{2} \begin{bmatrix} cos(A-B) + cos(A+B) \end{bmatrix}\]

  3. \[sinA cosB = \frac{1}{2} \begin{bmatrix} sin(A+B) + sin(A-B) \end{bmatrix}\]

  4. \[cosA sinB = \frac{1}{2} \begin{bmatrix} cos(A+B) + sin(A-B) \end{bmatrix}\]

 

Half Angle Identities:

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

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

  3. \[Tan(\frac{x}{2}) =  \pm \sqrt{\frac{1 - cos(x)}{1+ cos(x)}} = \frac{1 - cos(x)}{sin(x)} = \frac{sin(x)}{1 + cos(x)} \]


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. \[Sin \theta = \frac{e^{i\theta} - e^{-i\theta} }{2i}\]

  2. \[Cos \theta = \frac{e^{i\theta} + e^{-i\theta} }{2i}\]

  3. \[Tan \theta = \frac{e^{i\theta} - e^{-i\theta} }{e^{i\theta} + e^{-i\theta}}\]

  4. \[Cosec \theta = \frac{2i}{e^{i\theta} - e^{-i\theta} }\]

  5. \[Cosec \theta = \frac{2i}{e^{i\theta} + e^{-i\theta} }\]

  6. \[Cot \theta = \frac{e^{i\theta} + e^{-i\theta}}{e^{i\theta} - e^{-i\theta}}\]

 

Inverse Trigonometric Functions:

Definition:

  1. \[\theta = Sin^{-1}(x)\]  is equivalent to \[x = Sin \theta\]

  2. \[\theta = Cos^{-1}(x)\]  is equivalent to \[x = Cos \theta\]

  3. \[\theta = Tan^{-1}(x)\]  is equivalent to \[x = Tan \theta\]

 

Inverse Properties:

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

  1. \[sin(sin^{-1}(x)) = x\] 

  2. \[cos(cos^{-1}(x)) = x\] 

  3. \[tan(tan^{-1}(x)) = x\] 

  4. \[sin^{-1}(sin(\theta)) = \theta\] 

  5. \[cos^{-1}(cos(\theta)) = \theta\]

  6. \[tan^{-1}(tan(\theta)) = \theta\]

 

Other Notations

  1. \[sin^{-1}(x) = arcsin(x)\] 

  2. \[cos^{-1}(x) = arccos(x)\] 

  3. \[tan^{-1}(x) = arctan(x)\] 

 

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 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 \[Sin^{-1}(\frac{1}{\sqrt2})\]

Solution 

Let \[Sin^{-1}(\frac{1}{\sqrt2})\]  = α; 

then sin α = sin α = \[Sin^{-1}(\frac{1}{\sqrt2})\]  = sin 45 degree

α = 45 degree or π/4, which is the required principal value.


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

Solution

 L.H.S. =  \[Tan^{-1} \frac{1}{2} + Tan^{-1} \frac{1}{3}\]

= \[tan^{-1} \begin{pmatrix} \frac{\frac{1}{2} + \frac{1}{3}}{1 - \begin{pmatrix}  \frac{1}{2} \times \frac{1}{3}\end{pmatrix} } \end{pmatrix}  = tan^{-1} \begin{pmatrix} \frac{\frac{5}{6}}{\frac{5}{6}} \end{pmatrix}\]

 = \[tan^{-1}(1) = \frac{\pi}{4} \]


Fun Facts

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

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

  • Trigonometry is connected with music and architecture.


Conclusion

These are Trigonometry Formulas For Class 12 introduced in the Inverse trigonometric functions chapter of Class 12. Students can solve the problems based on these properties taking reference from this article. To get formulas for classes 10 and 11, students can visit Vedantu’s official website.


FAQs on Trigonometry Formulas For Class 12

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 the division operation between the other two. For example, tan θ = sin θ /cos θ and cot θ = cos θ / sin θ. The quotient that represents a trigonometric function by the division of a trigonometric function by another trigonometric function is called the quotient relation of trigonometric functions. In trigonometry, a trigonometric function can be divided by another trigonometric function and the quotient of them is also a trigonometric function. Hence, it is called the quotient identity of trigonometric functions.

2. Who is the father of trigonometry?

Trigonometry is the study of 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 the father of trigonometry.  12th Trigonometry formulas contain some trigonometry and their 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 was 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 helps solve 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 helps solve a triangle if all three sides, or two sides and the included angle are given.

4. How can students from class 12 get a list of Trigonometry Formulas For Class 12?

Students from class 12 can get a list of Trigonometry Formulas For Class 12 for free from Vedantu which have been prepared by the subject matter experts at Vedantu who have a vast knowledge of mathematics and understand the question paper pattern of class 12 maths board exam. Vedantu has an ample range of study materials for Class 12 maths such as formulas, important questions, sample papers, previous years’ questions, revision notes, and much more to help students from 12 boost their maths preparation level and get higher marks in class 12 maths board exam.

5. Why should students refer to Vedantu’s Trigonometry Formulas For Class 12?

Vedantu’s Trigonometry Formulas For Class 12 is a branch of mathematics that deals with angles, lengths and heights of triangles and relations between different parts of circles and other geometrical figures. Vedantu’s Trigonometry Formulas For Class 12 are accumulated by a panel of highly experienced mathematics experts after analyzing the previous years’ examination papers and study materials so that no important formula is left behind. This utmost precision in preparing these Trigonometry Formulas for the students for class 12 will ultimately lead to maximum marks in the class 12 maths board examination.