How are trigonometric identities helpful?

They make trigonometric expressions simpler, find solutions for equations, and assist in the proof of other mathematical relationships.

How are trigonometric identities proved?

Prove trigonometric identities by rearranging the left-hand side (LHS) to equal the right-hand side (RHS) using known formulae and algebraic methods.

Why do we use trigonometric derivatives?

Trigonometric derivatives are employed to determine the rate of change of trigonometric functions in calculus.

Why are trigonometric identities applied in geometry?

In geometry, trigonometric identities are used to solve angle problems, distance problems, and area problems, especially problems in right triangles, circles, and periodic functions.

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Trigonometric Ratios and Identities

Master Right Triangles & Equation Proofs in Minutes: Learn how to find unknown angles and sides, understand the foundational relationships of sine, cosine, and tangent, and master the step-by-step strategies to crack complex board exam proofs.

Class: 10 Mathematics (CBSE)
Chapter: Introduction to Trigonometry
Estimated Learning Time: 25–30 Minutes

1.0Learning Outcomes

After completing this lesson, you will be able to:

Define the six fundamental trigonometric ratios based on a right-angled triangle.
Recall and apply exact trigonometric values for standard angles (30, 45, and 60 degrees).
Master the reciprocal and quotient relationships.
State, derive, and manipulate the three primary Pythagorean identities.
Implement structured proving strategies to solve identity verification questions in board exams.

2.0Trigonometric Ratios

Six trigonometric ratios are defined as sine, cosine, tangent, and their reciprocal resp as cosecant, secant, and cotangent based on the ratio of any two sides in a right-angled triangle.

Here are the key Trigonometric Ratios:

Sine (sin): The sine of an angle within a right triangle is established As the ratio between the length of the side opposite to the angle and the length of the hypotenuse.

$sin (θ) = \frac{Opposite Side}{Hypotenuse} or sin θ = \frac{P}{H}$

Cosine (cos): The cosine of an angle within a right triangle is defined as the ratio between the length of the adjacent side of the angle and the hypotenuse length.

$cos (θ) = \frac{AdjacentSide}{Hypotenuse} or cos θ = \frac{B}{H}$

Tangent (tan): The tangent of an angle within a right triangle is defined as the ratio between the length of the opposite side of the angle and the length of the adjacent side.

$tan (θ) = \frac{Opposite Side}{Adjacent Side} or tan θ = \frac{P}{B}$

Cosecant (cosec): The cosecant of an angle within a right triangle is defined as the ratio between the hypotenuse length and the length of the side opposite the angle.

$cosec θ = \frac{Hypotenuse}{Opposite side} or cosec θ = \frac{H}{P}$

Secant (sec): The secant of an angle within a right triangle is defined as the ratio between the hypotenuse length and the length of the side adjacent to the angle.

$sec (θ) = \frac{Hypotenuse}{Adjacent Side} or sec θ = \frac{H}{B}$

Cotangent (cot): The cotangent of an angle within a right triangle is defined as the ratio between the adjacent side and the length of the side opposite the angle.

$cot (θ) = \frac{Adjacent Side}{Opposite Side} or cot θ = \frac{B}{P}$

These ratios are useful in solving various problems involving angles and distances in real-world applications such as Physics, Engineering, and Astronomy.

Trigonometric Identities

Trigonometric identities are fundamental tools for simplifying and solving problems related to angles in a right triangle. These formulas simplify complex equations, making them easier to handle in algebra, calculus, as well as geometry. Whether you're confirming an identity, simplifying expressions, or solving trigonometric equations, these trigonometric identities’ formulas form a solid base. This is why, here, we'll look into types of identities and how to use them.

Pythagorean Identities.
Reciprocal Identities.
Opposite Angle Identities
Complementary Angle Identities
Supplementary Angle Identities
Product Identities
Sum and Difference Identities
Sum to Product Identities
Double Angle Identities
Triple Angle Identities
Half Angle Identities

3.0Trigonometric Identities: Definition

A trigonometric identity can be defined as an equation that involves various trigonometric functions that are valid for all values of the involved variables as long as they are within the functions' domains. Trigonometric identities are basic mathematical tools, especially when dealing with angles and their connections. All of these trigonometric identities’ formulas are typically based on the six trigonometric ratios – sine, cosine, tangent, cosecant, secant, and cotangent.

4.0List of Trigonometric Identities

In trigonometry, there are a large number of trigonometric identities, each classified in different categories with a specific purpose. Every trigonometric identity is used not only for various problems but also for solving trigonometric identities themselves. Here is the list of some of the most important trigonometric identities:

Reciprocal Identities

Reciprocal identities are simply the relation between the six basic trigonometric functions mentioned above and their reciprocals. These include:

$sin θ = \frac{1}{c o s e c θ}$
$cos θ = \frac{1}{s e c θ}$
$tan θ = \frac{1}{c o t θ}$
$cot θ = \frac{1}{t a n θ}$
$cosec θ = \frac{1}{s i n θ}$
$sec θ = \frac{1}{c o s θ}$

Quotient Identities

These identities express the tangent and cotangent functions in terms of sine and cosine, like this:

$tan θ = \frac{s i n θ}{c o s θ}$
$cot θ = \frac{c o s θ}{s i n θ}$

Complementary and Supplementary Function Identities

As the name suggests, these identities relate the trigonometric functions to their complementary and supplementary angles. Note that two angles are complementary if their sum is 90, while two angles are supplementary if their sum equals 180.

Complementary Function Identities

$sin (90 - θ) = cos θ cos (90 - θ) = sin θ tan (90 - θ) = cot θ cosec (90 - θ) = sec θ sec (90 - θ) = cosec θ cot (90 - θ) = tan θ$

Supplementary Function Identities

$sin (180 - θ) = sin θ cos (180 - θ) = - cos θ tan (180 - θ) = - tan θ cosec (180 - θ) = cosec θ sec (180 - θ) = - sec θ cot (180 - θ) = - cot θ$

Trigonometric Identities for Sum and Difference of Any Two Angles

The formulas for the sum and difference enable us to determine the sine, cosine, and tangent of the sum or difference of two angles, say A and B. These are:

$sin (A + B) = sin A cos B + cos A sin B sin (A - B) = sin A cos B - cos A sin B cos (A + B) = cos A cos B - sin A sin B cos (A - B) = cos A cos B + sin A sin B tan (A + B) = \frac{tan A + tan B}{1 - tan A tan B} tan (A - B) = \frac{tan A - tan B}{1 + tan A tan B}$

Product–Sum Trigonometric Identities

These identities are the compound of the product and the sum of trigonometric ratios with different angles, say A and B.

$sin A + sin B = 2 sin (\frac{A + B}{2}) \cdot cos (\frac{A - B}{2}) cos A + cos B = 2 cos (\frac{A + B}{2}) \cdot cos (\frac{A - B}{2}) sin A - sin B = 2 cos (\frac{A + B}{2}) \cdot sin (\frac{A - B}{2}) cos A - cos B = - 2 sin (\frac{A + B}{2}) \cdot sin (\frac{A - B}{2})$

Trigonometric Identities of Products

$2 sin A sin B = cos (A - B) - cos (A + B) 2 sin A cos B = sin (A - B) - sin (A + B) 2 cos A cos B = cos (A + B) + cos (A + B)$

Pythagorean Identities

These are the most commonly used trigonometric identities derived from the Pythagorean theorem. These include:

$sin^{2} θ + cos^{2} θ = 1 1 + tan^{2} θ = sec^{2} θ 1 + cot^{2} θ = cosec^{2} θ$

Double Angle Trigonometric Identities

These are the identities used when the angle of trigonometric ratios is doubled, these identities are:

$sin 2 θ = 2 sin θ cos θ = \frac{2 tan θ}{1 + tan ^{2} θ} cos 2 θ = cos^{2} θ - sin^{2} θ = 2 cos^{2} θ - 1 = 1 - sin^{2} θ = \frac{1 - tan ^{2} θ}{1 + tan ^{2} θ} tan 2 θ = \frac{2 tan θ}{1 - tan θ}$

Half Angle Identities

These are the identities used when the angle of trigonometric ratios is halved; these identities are:

$sin (\frac{θ}{2}) = \pm \frac{1 - cos θ}{2} cos (\frac{θ}{2}) = \pm \frac{1 + cos θ}{2} tan (\frac{θ}{2}) = \pm \frac{1 - cos θ}{1 + cos θ}$

Triple Angle Identities

These are the identities used when the angle of trigonometric ratios are tripled, these identities are:

$sin 3 θ cos 3 θ tan 3 θ = 3 sin θ - 4 sin^{3} θ = 4 cos^{3} θ - 3 cos θ = \frac{3 tan θ - tan ^{3} θ}{1 - 3 tan ^{2} θ}$

5.0Proving Trigonometric Identities

In this section, we will focus mainly on proving the trigonometric identities of Pythagorean identity functions. These are the most important and frequently used identities, appearing in almost every complex problem of trigonometry.

To derive these identities, first, we will take a right-angled triangle ABC, as shown in the figure here:

Apply Pythagoras' theorem in triangle ABC, which states:

$A C^{2} = A B^{2} + B C^{2} \dots .. (a)$

For Pythagorean Identity 1:

Divide equation (a) by AC²; we will have:

$\frac{A C ^{2}}{A C ^{2}} = \frac{A B ^{2}}{A C ^{2}} + \frac{B C ^{2}}{A C ^{2}} \frac{A B ^{2}}{A C ^{2}} + \frac{B C ^{2}}{A C ^{2}} = 1 (\frac{A B}{A C})^{2} + (\frac{BC}{A C})^{2} = 1$

Since,

$sin θ = \frac{Perpendicular}{Hypotenuse} = \frac{BC}{A C} cos θ = \frac{Base}{Hypotenuse} = \frac{A B}{A C}$

Hence, equation 1 will become:

$(cos θ)^{2} + (sin θ)^{2} = 1 Or, sin^{2} θ + cos^{2} θ = 1 (cos θ)^{2} + (sin θ)^{2} = 1$

For Pythagorean Identity 2:

Divide the equation (a) by AB²

$\frac{A C ^{2}}{A B ^{2}} = \frac{A B ^{2}}{A B ^{2}} + \frac{B C ^{2}}{A B ^{2}} (\frac{A C}{A B})^{2} = 1 + (\frac{BC}{A C})^{2} \dots Since, tan θ = \frac{Perpendicular}{Base} = \frac{BC}{A B} sec θ = \frac{Hypotenuse}{Base} = \frac{A C}{A B}$

For Pythagorean Identity 3:

Divide equation (a) by BC²

$\frac{A C ^{2}}{B C ^{2}} = \frac{A B ^{2}}{B C ^{2}} + \frac{B C ^{2}}{B C ^{2}} (\frac{A C}{BC})^{2} = 1 + (\frac{A B}{BC})^{2} \dots Since, cosec θ = \frac{Hypotenuse}{Perpendicular} = \frac{A C}{BC} cot θ = \frac{Base}{Perpendicular} = \frac{A B}{BC}$

Hence, equation 3 will become,

$cosec^{2} θ = 1 + cot^{2} θ$

6.0Trigonometric Identities Derivatives

The derivatives of trigonometric identities are another set of important formulas in trigonometry as well as calculus. See the below-mentioned trigonometric identities table for these derivatives to understand them better:

Trigonometric Identities Derivatives
$\frac{d}{d x} [sin θ]$	$cos θ$
$\frac{d}{d x} [cos θ]$	$⊢ sin θ$
$\frac{d}{d x} [tan θ]$	$sec^{2} θ$
$\frac{d}{d x} [cot θ]$	$- cosec^{2} θ$
$\frac{d}{d x} [sec θ]$	$sec θ tan θ$
$\frac{d}{d x} [cosec θ]$	$- cosec θ cot θ$

7.0Solved Examples of Trigonometric Identities

Problem 1: If $sec θ + tan θ = p$ , then find the value of $sec θ - tan θ$ .

Solution: From Identity 2, we have:

$sec^{2} θ = 1 + tan^{2} θ sec^{2} θ - tan^{2} θ = 1 (sec θ + tan θ) (sec θ - tan θ) = 1 p (sec θ - tan θ) = 1 Hence, sec θ - tan θ = \frac{1}{p}$

Problem 2: Prove that: $\frac{1 - c o s ^{2} θ}{s i n θ} = sin θ$

Solution: From identity 1, we have:

$sin^{2} θ + cos^{2} θ = 1 sin^{2} θ = 1 - cos^{2} θ$

Taking the LHS of the equation

$\frac{1 - c o s ^{2} θ}{s i n θ}$

From equation 1:

$\frac{s i n ^{2} θ}{s i n θ} = sin θ$

LHS = RHS, hence proved.

Problem 3: Simplify the Expression $\frac{1 + t a n ^{2} θ}{1 + c o t ^{2} θ}$ .

Solution: From identities 2 and 3:

$1 + tan^{2} θ = sec^{2} θ 1 + cot^{2} θ = cosec^{2} θ$

Now, taking the comparison of the equation with these identities, we get:

$\frac{sec ^{2} θ}{cosec ^{2} θ} = \frac{\frac{1}{c o s ^{2} θ}}{\frac{1}{s i n ^{2} θ}} = \frac{sin ^{2} θ}{cos ^{2} θ} = (\frac{sin θ}{cos θ})^{2} \dots\dots (1)$

From quotient identities:

$tan θ = \frac{s i n θ}{c o s θ} \dots (2)$

From equations 1 and 2:

$\frac{1 + t a n ^{2} θ}{1 + c o t ^{2} θ} = tan^{2} θ$

8.0EUREKA by ALLEN – The Future of Class 10 Learning

EUREKA by ALLEN is designed to simplify, enrich, and enhance your experience in Class 10. Through the use of fun and engaging video lessons, regular practice tests, and immediate help for any doubts you may have regarding the material; students have a firm understanding of the concepts they are studying and feel confident in their preparation for their board exams. No matter if you are attempting to receive a higher mark or develop a better understanding of your studies, EUREKA will support you as you continue to grow as a learner.

Key Features of EUREKA Class 10 Courses:
Interactive concept learning modules Engaging story-based video lectures CBSE & State Board exam preparation Subjective answer writing practice Instant quizzes with performance feedback Detailed progress tracking 24×7 AI-powered doubt assistance NCERT-aligned curriculum Learn anytime, anywhere

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9.0Supporting Study Materials

This study material CBSE Notes and NCERT Solutions for the Chapter "Introduction to Trigonometry" on Trigonometric Ratios and Identities Topics is designed according to the latest CBSE Class 10 Mathematics syllabus and NCERT guidelines. It provides clear explanations of key concepts, definitions, formulas, and important questions to help students understand the six trigonometric ratios, trigonometric ratios of specific angles, standard trigonometric identities and prepare effectively for examinations.

CBSE Class 10 Maths Notes Chapter 8 Introduction To Trigonometry	NCERT Solutions for Class 10 Maths Chapter 8: Introduction to Trigonometry

10.030-Second Quick Revision

Trigonometric ratios show the relationship between sides and angles of a right-angled triangle.
The six main trigonometric ratios are sinθ, cosθ, tanθ, cotθ, secθ, and cosecθ.
sinθ = Perpendicular / Hypotenuse.
cosθ = Base / Hypotenuse.
tanθ = Perpendicular / Base.
Reciprocal relations: cosecθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ.
Important identity: sin²θ + cos²θ = 1.
Other identities: 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
Trigonometric identities help simplify and solve expressions.
Remember: SOH-CAH-TOA helps recall basic trigonometric ratios.

11.0Recommended Next Topics

Heights and Distances (Applications of Trigonometry)
Area Related to Circles (Sectors and Segments)
Arithmetic Progressions (AP)
Geometry: Properties of Tangents to a Circle

Trigonometric Ratios and Identities

Class: 10 Mathematics (CBSE)
Chapter: Introduction to Trigonometry
Estimated Learning Time: 25–30 Minutes

1.0Learning Outcomes

After completing this lesson, you will be able to:

Define the six fundamental trigonometric ratios based on a right-angled triangle.
Recall and apply exact trigonometric values for standard angles (30, 45, and 60 degrees).
Master the reciprocal and quotient relationships.
State, derive, and manipulate the three primary Pythagorean identities.
Implement structured proving strategies to solve identity verification questions in board exams.

2.0Trigonometric Ratios

Six trigonometric ratios are defined as sine, cosine, tangent, and their reciprocal resp as cosecant, secant, and cotangent based on the ratio of any two sides in a right-angled triangle.

Here are the key Trigonometric Ratios:

Sine (sin): The sine of an angle within a right triangle is established As the ratio between the length of the side opposite to the angle and the length of the hypotenuse.

$sin (θ) = \frac{Opposite Side}{Hypotenuse} or sin θ = \frac{P}{H}$

Cosine (cos): The cosine of an angle within a right triangle is defined as the ratio between the length of the adjacent side of the angle and the hypotenuse length.

$cos (θ) = \frac{AdjacentSide}{Hypotenuse} or cos θ = \frac{B}{H}$

Tangent (tan): The tangent of an angle within a right triangle is defined as the ratio between the length of the opposite side of the angle and the length of the adjacent side.

$tan (θ) = \frac{Opposite Side}{Adjacent Side} or tan θ = \frac{P}{B}$

Cosecant (cosec): The cosecant of an angle within a right triangle is defined as the ratio between the hypotenuse length and the length of the side opposite the angle.

$cosec θ = \frac{Hypotenuse}{Opposite side} or cosec θ = \frac{H}{P}$

Secant (sec): The secant of an angle within a right triangle is defined as the ratio between the hypotenuse length and the length of the side adjacent to the angle.

$sec (θ) = \frac{Hypotenuse}{Adjacent Side} or sec θ = \frac{H}{B}$

Cotangent (cot): The cotangent of an angle within a right triangle is defined as the ratio between the adjacent side and the length of the side opposite the angle.

$cot (θ) = \frac{Adjacent Side}{Opposite Side} or cot θ = \frac{B}{P}$

These ratios are useful in solving various problems involving angles and distances in real-world applications such as Physics, Engineering, and Astronomy.

Trigonometric Identities

Pythagorean Identities.
Reciprocal Identities.
Opposite Angle Identities
Complementary Angle Identities
Supplementary Angle Identities
Product Identities
Sum and Difference Identities
Sum to Product Identities
Double Angle Identities
Triple Angle Identities
Half Angle Identities

3.0Trigonometric Identities: Definition

4.0List of Trigonometric Identities

Reciprocal Identities

Reciprocal identities are simply the relation between the six basic trigonometric functions mentioned above and their reciprocals. These include:

$sin θ = \frac{1}{c o s e c θ}$
$cos θ = \frac{1}{s e c θ}$
$tan θ = \frac{1}{c o t θ}$
$cot θ = \frac{1}{t a n θ}$
$cosec θ = \frac{1}{s i n θ}$
$sec θ = \frac{1}{c o s θ}$

Quotient Identities

These identities express the tangent and cotangent functions in terms of sine and cosine, like this:

$tan θ = \frac{s i n θ}{c o s θ}$
$cot θ = \frac{c o s θ}{s i n θ}$

Complementary and Supplementary Function Identities

Complementary Function Identities

$sin (90 - θ) = cos θ cos (90 - θ) = sin θ tan (90 - θ) = cot θ cosec (90 - θ) = sec θ sec (90 - θ) = cosec θ cot (90 - θ) = tan θ$

Supplementary Function Identities

$sin (180 - θ) = sin θ cos (180 - θ) = - cos θ tan (180 - θ) = - tan θ cosec (180 - θ) = cosec θ sec (180 - θ) = - sec θ cot (180 - θ) = - cot θ$

Trigonometric Identities for Sum and Difference of Any Two Angles

The formulas for the sum and difference enable us to determine the sine, cosine, and tangent of the sum or difference of two angles, say A and B. These are:

Product–Sum Trigonometric Identities

These identities are the compound of the product and the sum of trigonometric ratios with different angles, say A and B.

Trigonometric Identities of Products

$2 sin A sin B = cos (A - B) - cos (A + B) 2 sin A cos B = sin (A - B) - sin (A + B) 2 cos A cos B = cos (A + B) + cos (A + B)$

Pythagorean Identities

These are the most commonly used trigonometric identities derived from the Pythagorean theorem. These include:

$sin^{2} θ + cos^{2} θ = 1 1 + tan^{2} θ = sec^{2} θ 1 + cot^{2} θ = cosec^{2} θ$

Double Angle Trigonometric Identities

These are the identities used when the angle of trigonometric ratios is doubled, these identities are:

Half Angle Identities

These are the identities used when the angle of trigonometric ratios is halved; these identities are:

$sin (\frac{θ}{2}) = \pm \frac{1 - cos θ}{2} cos (\frac{θ}{2}) = \pm \frac{1 + cos θ}{2} tan (\frac{θ}{2}) = \pm \frac{1 - cos θ}{1 + cos θ}$

Triple Angle Identities

These are the identities used when the angle of trigonometric ratios are tripled, these identities are:

$sin 3 θ cos 3 θ tan 3 θ = 3 sin θ - 4 sin^{3} θ = 4 cos^{3} θ - 3 cos θ = \frac{3 tan θ - tan ^{3} θ}{1 - 3 tan ^{2} θ}$

5.0Proving Trigonometric Identities

To derive these identities, first, we will take a right-angled triangle ABC, as shown in the figure here:

Apply Pythagoras' theorem in triangle ABC, which states:

$A C^{2} = A B^{2} + B C^{2} \dots .. (a)$

For Pythagorean Identity 1:

Divide equation (a) by AC²; we will have:

$\frac{A C ^{2}}{A C ^{2}} = \frac{A B ^{2}}{A C ^{2}} + \frac{B C ^{2}}{A C ^{2}} \frac{A B ^{2}}{A C ^{2}} + \frac{B C ^{2}}{A C ^{2}} = 1 (\frac{A B}{A C})^{2} + (\frac{BC}{A C})^{2} = 1$

Since,

$sin θ = \frac{Perpendicular}{Hypotenuse} = \frac{BC}{A C} cos θ = \frac{Base}{Hypotenuse} = \frac{A B}{A C}$

Hence, equation 1 will become:

$(cos θ)^{2} + (sin θ)^{2} = 1 Or, sin^{2} θ + cos^{2} θ = 1 (cos θ)^{2} + (sin θ)^{2} = 1$

For Pythagorean Identity 2:

Divide the equation (a) by AB²

For Pythagorean Identity 3:

Divide equation (a) by BC²

Hence, equation 3 will become,

$cosec^{2} θ = 1 + cot^{2} θ$

6.0Trigonometric Identities Derivatives

Trigonometric Identities Derivatives
$\frac{d}{d x} [sin θ]$	$cos θ$
$\frac{d}{d x} [cos θ]$	$⊢ sin θ$
$\frac{d}{d x} [tan θ]$	$sec^{2} θ$
$\frac{d}{d x} [cot θ]$	$- cosec^{2} θ$
$\frac{d}{d x} [sec θ]$	$sec θ tan θ$
$\frac{d}{d x} [cosec θ]$	$- cosec θ cot θ$

7.0Solved Examples of Trigonometric Identities

Problem 1: If $sec θ + tan θ = p$ , then find the value of $sec θ - tan θ$ .

Solution: From Identity 2, we have:

$sec^{2} θ = 1 + tan^{2} θ sec^{2} θ - tan^{2} θ = 1 (sec θ + tan θ) (sec θ - tan θ) = 1 p (sec θ - tan θ) = 1 Hence, sec θ - tan θ = \frac{1}{p}$

Problem 2: Prove that: $\frac{1 - c o s ^{2} θ}{s i n θ} = sin θ$

Solution: From identity 1, we have:

$sin^{2} θ + cos^{2} θ = 1 sin^{2} θ = 1 - cos^{2} θ$

Taking the LHS of the equation

$\frac{1 - c o s ^{2} θ}{s i n θ}$

From equation 1:

$\frac{s i n ^{2} θ}{s i n θ} = sin θ$

LHS = RHS, hence proved.

Problem 3: Simplify the Expression $\frac{1 + t a n ^{2} θ}{1 + c o t ^{2} θ}$ .

Solution: From identities 2 and 3:

$1 + tan^{2} θ = sec^{2} θ 1 + cot^{2} θ = cosec^{2} θ$

Now, taking the comparison of the equation with these identities, we get:

$\frac{sec ^{2} θ}{cosec ^{2} θ} = \frac{\frac{1}{c o s ^{2} θ}}{\frac{1}{s i n ^{2} θ}} = \frac{sin ^{2} θ}{cos ^{2} θ} = (\frac{sin θ}{cos θ})^{2} \dots\dots (1)$

From quotient identities:

$tan θ = \frac{s i n θ}{c o s θ} \dots (2)$

From equations 1 and 2:

$\frac{1 + t a n ^{2} θ}{1 + c o t ^{2} θ} = tan^{2} θ$

8.0EUREKA by ALLEN – The Future of Class 10 Learning

Key Features of EUREKA Class 10 Courses:
Interactive concept learning modules Engaging story-based video lectures CBSE & State Board exam preparation Subjective answer writing practice Instant quizzes with performance feedback Detailed progress tracking 24×7 AI-powered doubt assistance NCERT-aligned curriculum Learn anytime, anywhere

Explore Now

9.0Supporting Study Materials

CBSE Class 10 Maths Notes Chapter 8 Introduction To Trigonometry	NCERT Solutions for Class 10 Maths Chapter 8: Introduction to Trigonometry

10.030-Second Quick Revision

Trigonometric ratios show the relationship between sides and angles of a right-angled triangle.
The six main trigonometric ratios are sinθ, cosθ, tanθ, cotθ, secθ, and cosecθ.
sinθ = Perpendicular / Hypotenuse.
cosθ = Base / Hypotenuse.
tanθ = Perpendicular / Base.
Reciprocal relations: cosecθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ.
Important identity: sin²θ + cos²θ = 1.
Other identities: 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
Trigonometric identities help simplify and solve expressions.
Remember: SOH-CAH-TOA helps recall basic trigonometric ratios.

11.0Recommended Next Topics

Heights and Distances (Applications of Trigonometry)
Area Related to Circles (Sectors and Segments)
Arithmetic Progressions (AP)
Geometry: Properties of Tangents to a Circle

Frequently Asked Questions

How are trigonometric identities helpful?

How are trigonometric identities proved?

Why do we use trigonometric derivatives?

Why are trigonometric identities applied in geometry?

Frequently Asked Questions

How are trigonometric identities helpful?

How are trigonometric identities proved?

Why do we use trigonometric derivatives?

Why are trigonometric identities applied in geometry?

Join ALLEN!

Trigonometric Ratios and Identities

1.0Learning Outcomes

2.0Trigonometric Ratios

Trigonometric Identities

3.0Trigonometric Identities: Definition

4.0List of Trigonometric Identities

Reciprocal Identities

Quotient Identities

Complementary and Supplementary Function Identities

Complementary Function Identities

Supplementary Function Identities

Trigonometric Identities for Sum and Difference of Any Two Angles

Product–Sum Trigonometric Identities

Trigonometric Identities of Products

Pythagorean Identities

Double Angle Trigonometric Identities

Half Angle Identities

Triple Angle Identities

5.0Proving Trigonometric Identities

6.0Trigonometric Identities Derivatives

7.0Solved Examples of Trigonometric Identities

8.0EUREKA by ALLEN – The Future of Class 10 Learning

9.0Supporting Study Materials

10.030-Second Quick Revision

11.0Recommended Next Topics

On this page

Join ALLEN!

Trigonometric Ratios and Identities

1.0Learning Outcomes

2.0Trigonometric Ratios

Trigonometric Identities

3.0Trigonometric Identities: Definition

4.0List of Trigonometric Identities

Reciprocal Identities

Quotient Identities

Complementary and Supplementary Function Identities

Complementary Function Identities

Supplementary Function Identities

Trigonometric Identities for Sum and Difference of Any Two Angles

Product–Sum Trigonometric Identities

Trigonometric Identities of Products

Pythagorean Identities

Double Angle Trigonometric Identities

Half Angle Identities

Triple Angle Identities

5.0Proving Trigonometric Identities

6.0Trigonometric Identities Derivatives

7.0Solved Examples of Trigonometric Identities

8.0EUREKA by ALLEN – The Future of Class 10 Learning

9.0Supporting Study Materials

10.030-Second Quick Revision

11.0Recommended Next Topics

On this page