Trigonometry is a branch of mathematics that studies the relationships between angles and the sides of triangles, especially right triangles. It plays a key role in various fields such as physics, engineering, astronomy, and computer graphics. Understanding trigonometry formulas and identities is essential for solving a wide range of problems in these domains.
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Here are some important types of trigonometry formulas and identities:
1. Basic Trigonometric Functions
The primary trigonometric functions are defined using a right triangle:
 Sine (sin): $sin(θ)=hypotenuseopposite $
 Cosine (cos): $cos(θ)=hypotenuseadjacent $
 Tangent (tan): $tan(θ)=cos(θ)sin(θ) =adjacentopposite $
2. Pythagorean Theorem
The Pythagorean theorem is fundamental to trigonometry:
$a_{2}+b_{2}=c_{2}$
Where \( a \) and \( b \) are the lengths of the legs of a right triangle, and \( c \) is the length of the hypotenuse.
3. Pythagorean Identities
These identities are derived from the Pythagorean theorem and involve the square of the trigonometric functions:
 $_{2}(θ)+_{2}(θ)=1$
 $1+_{2}(θ)=_{2}(θ)$
 $1+_{2}(θ)=_{2}(θ)$
4. Reciprocal Identities
These express one trigonometric function in terms of another:
 $sec(θ)=cos(θ)1 $
 $csc(θ)=sin(θ)1 $
 $cot(θ)=tan(θ)1 $
5. Angle Sum and Difference Identities
These formulas relate the trigonometric functions of sums and differences of angles:
 Sine: $sin(α+β)=sin(α)cos(β)+cos(α)sin(β)$
 Cosine: $cos(α+β)=cos(α)cos(β)−sin(α)sin(β)$
 Tangent: $tan(α+β)=−tan(α)tan(β)tan(α)+tan(β) $
6. Double Angle Identities
These identities express trigonometric functions at double angles:
 Sine: $sin(2θ)=2sin(θ)cos(θ)$
 Cosine: $cos(2θ)=_{2}(θ)−_{2}(θ)$
 Tangent: $tan(2θ)=−tan(θ)tan(θ) $
7. Half Angle Identities
These identities express trigonometric functions at half angles:
 Sine: $sin(2θ )=±2−cos(θ) $
 Cosine: $cos(2θ )=±2+cos(θ) $
 Tangent: $tan(2θ )=±+cos(θ)−cos(θ) $
8. Law of Sines and Law of Cosines
These laws relate the sides and angles of any triangle, not just right triangles:
 Law of Sines: $asin(A) =bsin(B) =csin(C) $
 Law of Cosines: $c_{2}=a_{2}+b_{2}−2abcos(C)$
These laws are essential for solving nonright triangles.
9. Unit Circle
The unit circle is a graphical representation of trigonometric functions. It is a circle with a radius of one, centered at the origin of a coordinate system. Trigonometric functions can be derived from the unit circle:
– The angle in radians around the circle is measured counterclockwise.
– Points on the circle represent the sine and cosine values for angles from 0 to \(2\pi\).
Understanding these formulas and identities helps you solve trigonometric problems and equations efficiently. Practice problems are useful for reinforcing your knowledge and improving your skills.
Trigonometry is a branch of mathematics that focuses on studying the relationships between angles and lengths in triangles. It originated from geometric problems involving right triangles but has since expanded to encompass various shapes and periodic phenomena. Trigonometry is built on the fundamental trigonometric functions—sine, cosine, and tangent—which describe the ratios of certain side lengths in right triangles to their angles.
These functions form the basis for understanding and solving problems involving angles, distances, and heights in various fields such as physics, engineering, architecture, and astronomy. For example, trigonometry is used to calculate the height of buildings, the distance between celestial objects, and the behavior of waves.
Additionally, trigonometry plays a crucial role in understanding periodic functions and patterns, as well as in the analysis of sound and light waves. Trigonometric identities and formulas, such as the Pythagorean identity, are essential tools for simplifying and solving complex mathematical problems. Overall, trigonometry is a versatile and essential field of mathematics that underpins many scientific and engineering applications.
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 
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 
Trigonometry Ratios Table 
Trigonometry Functions
Trigonometric functions are mathematical tools that relate angles in right triangles to the lengths of the sides. These functions have wideranging applications across physics, engineering, astronomy, and other scientific fields.
The primary trigonometric functions include:
 Sine (sin): Relates the length of the side opposite an angle to the length of the hypotenuse in a right triangle.
 Cosine (cos): Relates the length of the adjacent side to the length of the hypotenuse.
 Tangent (tan): Relates the length of the side opposite an angle to the length of the adjacent side.
The secondary functions include:
 Cotangent (cot): The reciprocal of tangent.
 Secant (sec): The reciprocal of cosine.
 Cosecant (csc): The reciprocal of sine.
These functions are essential for solving problems involving angles, distances, and waveforms. They also help in modeling periodic phenomena and analyzing sound and light waves. Trigonometric functions are foundational in various mathematical and practical applications.
Trigonometric Function  Domain  Range  Period 

sin(θ)  All Real Number i.e., R  [1, 1]  2π or 360° 
cos(θ)  All Real Numbers i.e.,  [1, 1]  2π or 360° 
tan(θ)  All Real Numbers excluding odd multiples of π/2  R  π or 180° 
cot(θ)  All Real Numbers excluding multiples of π  R  2π or 360° 
sec(θ)  All Real Numbers excluding values where cos(x) = 0  R[1, 1]  2π or 360° 
cosec(θ)  All Real Numbers excluding multiples of π  R[1, 1]  π or 180° 
Trigonometry Formula Overview
Trigonometry is a branch of mathematics that studies the relationships between angles and the sides of triangles, particularly rightangled triangles. In a rightangled triangle, one of the angles is a right angle (90 degrees), which gives rise to specific relationships between the three sides of the triangle.
The three sides of a rightangled triangle are:
 Hypotenuse: This is the longest side of the triangle and is opposite the right angle. It represents the diagonal distance between the two other sides.
 Opposite Side (Perpendicular): This side is opposite to the angle in question (not the right angle). It forms the other part of the right angle.
 Adjacent Side (Base): This side is adjacent to the angle in question and is not the hypotenuse. It forms the base of the triangle along with the hypotenuse.
These sides and their relationships with the angles in the triangle give rise to the fundamental trigonometric functions: sine, cosine, and tangent.
 Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse. $sin(θ)=hypotenuseopposite side $
 Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse. $cos(θ)=hypotenuseadjacent side $
 Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side. $tan(θ)=adjacent sideopposite side $
The inverse functions (arcsine, arccosine, and arctangent) can be used to find the angles when the ratios are known.
Understanding the relationships between the sides and angles of a rightangled triangle is fundamental to trigonometry. These principles can be extended to other geometric shapes and contexts, providing the basis for more advanced mathematical, engineering, and physical applications.
Trigonometry is a branch of mathematics that studies the relationships between angles and the sides of triangles, particularly rightangled triangles. This area of mathematics provides a wide range of formulas and identities that are critical for solving problems in various mathematical and scientific fields. Below, you’ll find an overview of the different types of trigonometry formulas and ratios that are essential for students in Classes 9, 10, 11, and 12.
Here is the list of formulas in trigonometry we are going to discuss:
 Basic Trigonometric Ratio Formulas
 Unit Circle Formulas
 Trigonometric Identities
Basic Trigonometric Ratios
In trigonometry, there are six fundamental functions known as trigonometric ratios. These ratios describe the relationships between the angles and sides of a right triangle. The six trigonometric functions are:
 Sine (sin): The ratio of the length of the side opposite an angle to the length of the hypotenuse.
 Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse.
 Tangent (tan): The ratio of the length of the side opposite an angle to the length of the adjacent side.
 Cotangent (cot): The reciprocal of tangent, it relates the length of the adjacent side to the length of the side opposite an angle.
 Secant (sec): The reciprocal of cosine, it relates the length of the hypotenuse to the length of the adjacent side.
 Cosecant (csc): The reciprocal of sine, it relates the length of the hypotenuse to the length of the side opposite an angle.
These trigonometric ratios are essential in solving problems involving angles, lengths, and periodic functions, and they have extensive applications in various fields such as physics, engineering, and mathematics.
List of Trigonometric Ratios 


Trigonometric Ratio  Definition 
sin θ  Perpendicular / Hypotenuse 
cos θ  Base / Hypotenuse 
tan θ  Perpendicular / Base 
sec θ  Hypotenuse / Base 
cosec θ  Hypotenuse / Perpendicular 
cot θ  Base / Perpendicular 
Unit Circle Formula in Trigonometry
The unit circle is a fundamental concept in trigonometry that serves as the basis for understanding and calculating trigonometric functions for all angles. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system. Here’s how trigonometric functions are derived from the unit circle and how they are related to the coordinates of a point on the circle.
In the unit circle, an angle $θ$ is measured counterclockwise from the positive xaxis. As the angle increases, the point on the unit circle corresponding to the angle moves along the circle.
Given a point $(x,y)$ on the unit circle, where the xcoordinate represents the horizontal distance from the origin and the ycoordinate represents the vertical distance from the origin, the following trigonometric functions can be defined:
 Sine (sin): The sine of angle $θ$ is equal to the ycoordinate of the point on the unit circle corresponding to angle $θ$. $sin(θ)=y$
 Cosine (cos): The cosine of angle $θ$ is equal to the xcoordinate of the point on the unit circle corresponding to angle $θ$. $cos(θ)=x$
 Tangent (tan): The tangent of angle $θ$ is the ratio of the sine of $θ$ to the cosine of $θ$. It can also be interpreted as the slope of the line from the origin to the point on the unit circle. $tan(θ)=xy $
 Cotangent (cot): The cotangent of angle $θ$ is the reciprocal of the tangent of $θ$. $cot(θ)=yx $
 Secant (sec): The secant of angle $θ$ is the reciprocal of the cosine of $θ$. $sec(θ)=x1 $
 Cosecant (csc): The cosecant of angle $θ$ is the reciprocal of the sine of $θ$. $csc(θ)=y1 $
These functions and their relationships form the basis of trigonometry and can be used to calculate the sides and angles of rightangled triangles. The unit circle provides a convenient way to visualize and understand the behavior of trigonometric functions across different angles and quadrants.
Trigonometric Identities
Trigonometric identities are equations that express relationships between trigonometric functions. They are universally true for all values of the variables within their domains. These identities are essential for simplifying complex expressions, solving equations, and understanding the behavior of trigonometric functions.
 Reciprocal Identities
 Pythagorean Identities
 Periodicity Identities (in Radians)
 Even and Odd Angle Formula
 Cofunction identities (in Degrees)
 Sum and Difference Identities
 Double Angle Identities
 Inverse Trigonometry Formulas
 Triple Angle Identities
 Half Angle Identities
 Sum to Product Identities
 Product Identities
Let’s discuss these identites in detail.
Reciprocal Identities
eciprocal identities express each of the primary trigonometric functions in terms of its reciprocal. They are derived from the definition of the basic trigonometric functions and provide alternate forms for the functions:
 Cosecant (csc): The reciprocal of sine. $csc(θ)=sin(θ)1 $
 Secant (sec): The reciprocal of cosine. $sec(θ)=cos(θ)1 $
 Cotangent (cot): The reciprocal of tangent. $cot(θ)=tan(θ)1 $
Likewise, the reciprocal identities can also be expressed in terms of the primary functions:
 $sin(θ)=csc(θ)1 $
 $cos(θ)=sec(θ)1 $
 $tan(θ)=cot(θ)1 $
Pythagorean Identities
Pythagorean identities are derived from the Pythagorean theorem and express relationships between the squares of trigonometric functions:
 $_{2}(θ)+_{2}(θ)=1$
 $1+_{2}(θ)=_{2}(θ)$
 $1+_{2}(θ)=_{2}(θ)$
These identities are useful for converting between trigonometric functions and simplifying complex expressions.
Periodicity Identities (in Radians)
Periodicity identities express the cyclic nature of trigonometric functions. These identities indicate how trigonometric functions repeat their values after a certain interval:
All trigonometric identities repeat themselves after a particular period. Hence are cyclic in nature. This period for the repetition of values is different for different trigonometric 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
Here’s a table that compares the trigonometric properties in different quadrants :
Quadrant  Sine (sin θ)  Cosine (cos θ)  Tangent (tan θ)  Cosecant (csc θ)  Secant (sec θ)  Cotangent (cot θ) 

I (0° to 90°)  Positive  Positive  Positive  Positive  Positive  Positive 
II (90° to 180°)  Positive  Negative  Negative  Positive  Negative  Negative 
III (180° to 270°)  Negative  Negative  Positive  Negative  Negative  Positive 
IV (270° to 360°)  Negative  Positive  Negative  Negative  Positive  Negative 
Even and Odd Angle Formula
Trigonometric functions can be classified as even or odd, depending on whether their signs change with the sign of the angle:
 Even Function: Cosine and secant are even functions, which means they remain unchanged when the angle is negated.
 $cos(−θ)=cos(θ)$
 $sec(−θ)=sec(θ)$
 Odd Function: Sine, cosecant, tangent, and cotangent are odd functions, which means their signs reverse when the angle is negated.
 $sin(−θ)=−sin(θ)$
 $csc(−θ)=−csc(θ)$
 $tan(−θ)=−tan(θ)$
 $cot(−θ)=−cot(θ)$
These formulas are useful for simplifying trigonometric expressions and solving equations involving negative angles.
Cofunction identities (in Degrees)
Cofunction identities express the relationship between trigonometric functions of complementary angles (90° – θ). These identities allow us to relate sine to cosine, tangent to cotangent, and secant to cosecant:
sin(90°  x) = cos x
andcos(90°  x) = sin x
tan(90°  x) = cot x
andcot(90°  x) = tan x
sec(90°  x) = csc x
andcosec(90°  x) = sec x
Sum and Difference Identities
Sum and difference identities relate the sine, cosine, and tangent of the sum or difference of two angles to the trigonometric functions of the individual angles:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
andsin(x  y) = sin(x)cos(y)  cos(x)sin(y)
cos(x + y) = cos(x)cos(y)  sin(x)sin(y)
andcos(x  y) = cos(x)cos(y) + sin(x)sin(y)
tan(x + y) = (tan(x) + tan(y)) / (1  tan(x)tan(y))
andtan(x  y) = (tan(x)  tan(y)) / (1 + tan(x)tan(y))
Double Angle Identities
Double angle identities express trigonometric functions of double the angle (2θ) in terms of the original angle (θ):
 sin (2x) = 2sin(x) • cos(x) = [2tan x/(1 + tan^{2} x)]
 cos (2x) = cos^{2}(x) – sin^{2}(x) = [(1 – tan^{2} x)/(1 + tan^{2} x)] = 2cos^{2}(x) – 1 = 1 – 2sin^{2}(x)
 tan (2x) = [2tan(x)]/ [1 – tan^{2}(x)]
 sec (2x) = sec^{2 }x/(2 – sec^{2} x)
 cosec (2x) = (sec x • cosec x)/2
Inverse Trigonometry Formulas
Inverse trigonometry formulas involve the inverses of the basic trigonometric functions and are used to find the angle that corresponds to a given trigonometric ratio:
 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
Triple Angle Identities
Triple angle identities express trigonometric functions of triple the angle (3θ) in terms of the original angle (θ):
sin 3x=3sin x – 4sin^{3}x
cos 3x=4cos^{3}x – 3cos x
$tan3x=−tanxtanx−tanx $
Half Angle Identities
Half angle identities express trigonometric functions of half an angle (θ/2) in terms of the original angle:
$sin2x =±2−cosx $
$cos2x =±2+cosx $
$tan(2x )=±+cos(x)−cos(x) $
Also,
$tan(2x )=±+cos(x)−cos(x) $
$tan(2x )=±(+cos(x))(−cos(x))(−cos(x))(−cos(x)) $
$=−cos(x)(−cos(x))2 $
$=sin(x)(−cos(x))2 $
$=sin(x)−cos(x) $
$tan(2x )=sin(x)−cos(x) $
Sum to Product Identities
Sum to product identities express sums or differences of trigonometric functions as products:
 sinx + siny = 2[sin((x + y)/2)cos((x − y)/2)]
 sinx − siny = 2[cos((x + y)/2)sin((x − y)/2)]
 cosx + cosy = 2[cos((x + y)/2)cos((x − y)/2)]
 cosx − cosy = −2[sin((x + y)/2)sin((x − y)/2)]
Product Identities
Product identities express products of trigonometric functions as sums or differences:
These trigonometric formulas are derived from the sum and difference formulas for sine and cosine.
 sinx⋅cosy = [sin(x + y) + sin(x − y)]/2
 cosx⋅cosy = [cos(x + y) + cos(x − y)]/2
 sinx⋅siny = [cos(x − y) − cos(x + y)]/2
List of Trigonometry Formulas
The table given below consists of basic trigonometry ratios for angles such as such as 0°, 30°, 45°, 60°, and 90° that are commonly used for solving problems.
Table of Trigonometric Ratios Advertisements



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 
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 
Solved Questions on Trigonometry Formula:
Here are 9 solved questions on trigonometry formulas to help you get a better grasp of the concepts:
Question 1
Given: $tanα+secα=2$
Find: $tanα−secα$
Solution:
Given $tanα+secα=2$
To find $tanα−secα$, use the formula:
 $(tanα+secα)×(tanα−secα)=_{2}α−_{2}α$
Since $_{2}α−_{2}α=1$, we have:
 $(2)(tanα−secα)=1$
So, $tanα−secα=21 $
Question 2
Given: $sinβ=54 $
Find: $cosβ$
Solution:
Using Pythagorean identity: $_{2}β+_{2}β=1$
Given $sinβ=54 $, so:
 $(54 )_{2}+_{2}β=1$
 $2516 +_{2}β=1$
 $_{2}β=259 $
Therefore, $cosβ=53 $
Question 3
Given: $sinγ=53 $
Find: $tanγ$
Solution:
Given $sinγ=53 $
From Pythagorean identity: $cosγ=−sin2γ $
So, $cosγ=−(53 ) $
 $cosγ=2516 =54 $
Then, use the definition of $tanγ$:
 $tanγ=cosγsinγ $
 $tanγ=53 ÷54 =43 $
Therefore, $tanγ=43 $
Question 4
Given: $cos(2θ)$
Find: $_{2}θ−_{2}θ$
Solution:
Use the double angle formula:
 $cos(2θ)=_{2}θ−_{2}θ$
Hence, $_{2}θ−_{2}θ=cos(2θ)$
Question 5
Given: $sin(4_{∘})$
Find: $cos(4_{∘})$
Solution:
Both $sin(4_{∘})$ and $cos(4_{∘})$ are equal:
 $sin(4_{∘})=cos(4_{∘})=22 $
Therefore, $cos(4_{∘})=22 $
Question 6
Given: $cos(3_{∘})$
Find: $tan(3_{∘})$
Solution:
Given $cos(3_{∘})=23 $ and $sin(3_{∘})=21 $, then:
 $tan(3_{∘})=cos()sin() $
 $tan(3_{∘})=21 ÷23 =3 1 $
Therefore, $tan(3_{∘})=3 1 $
Question 7
Given: $sinϕ=cos(9_{∘}−ϕ)$
Prove: $cosϕ=sin(9_{∘}−ϕ)$
Solution:
Given $sinϕ=cos(9_{∘}−ϕ)$
By definition:
 $cos(9_{∘}−ϕ)=sinϕ$
 Therefore, $cosϕ=sin(9_{∘}−ϕ)$
Hence, the statement is proven.
Question 8
Given: $_{2}θ+_{2}θ=1$
Find: $_{2}θ+1=_{2}θ$
Solution:
Using the Pythagorean identity:
 $_{2}θ+_{2}θ=1$
 Divide each side by $_{2}θ$:
 $_{2}θ+1=_{2}θ$
Hence, the identity is proven.
Question 9
Given: $_{2}x−_{2}x=cos(2x)$
Find: Prove that $2sinxcosx=sin(2x)$
Solution:
Given $_{2}x−_{2}x=cos(2x)$, from the double angle formula, $_{2}x−_{2}x=cos(2x)$.
Now, prove the other double angle identity:
 $2sinxcosx=sin(2x)$
Given:
 $sin(2x)=2sinxcosx$
Hence, $2sinxcosx=sin(2x)$.