Trigonometric Ratios | Definitions, Formulas & Identities, Solved Examples

ANAND CLASSES Study Material and Notes to learn Class 10 Trigonometric Ratios with clear definitions, formulas, and identities. Perfect for CBSE Class 10 & competitive exams JEE, NDA.

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📚 Basic Concepts of Trigonometry

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🔹 What is Trigonometry?

Trigonometry is the branch of mathematics that deals with relationships between angles and sides of a triangle.

The word comes from:

• ‘tri’ – meaning three
• ‘gon’ – meaning sides
• ‘metron’ – meaning measure

It originated from the need to compute distances in astronomy and is now widely used in physics, engineering, and navigation.

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🔹 What are Trigonometric Ratios?

Trigonometric ratios relate the angles of a right triangle to the lengths of its sides.

Consider a right triangle ABC, right-angled at B:

• AC: hypotenuse
• BC: side opposite to angle A (called perpendicular)
• AB: side adjacent to angle A (called base)

The six trigonometric ratios for angle A are defined as:

$$\sin A = \frac{\text{side opposite to angle } A}{\text{hypotenuse}} = \frac{BC}{AC}$$

$$\cos A = \frac{\text{side adjacent to angle } A}{\text{hypotenuse}} = \frac{AB}{AC} $$

$$\tan A = \frac{\text{side opposite to angle } A}{\text{side adjacent to angle } A} = \frac{BC}{AB} $$

$$\cot A = \frac{\text{side adjacent to angle } A}{\text{side opposite to angle } A} = \frac{AB}{BC}$$

$$\sec A = \frac{\text{hypotenuse}}{\text{side adjacent to angle } A} = \frac{AC}{AB}$$

$$\mathrm{cosec}\;A = \frac{\text{hypotenuse}}{\text{side opposite to angle } A} = \frac{AC}{BC}$$

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🔄 Relationship between Trigonometric Ratios

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🔸 Reciprocal Identities

Each ratio has a reciprocal:

Identity.1 :

$$\sin A \cdot \mathrm{cosec}\;A = \frac{BC}{AC} \cdot \frac{AC}{BC} = 1 \quad $$

$$ \quad \sin A = \frac{1}{\mathrm{cosec}\;A}, \quad \mathrm{cosec}\;A = \frac{1}{\sin A}$$

Identity.2 :

$$\cos A \cdot \sec A = \frac{AB}{AC} \cdot \frac{AC}{AB} = 1 \quad $$

$$\quad \cos A = \frac{1}{\sec A}, \quad \sec A = \frac{1}{\cos A}$$

Identity.3 :

$$\tan A \cdot \cot A = \frac{BC}{AB} \cdot \frac{AB}{BC} = 1 \quad$$

$$ \quad \tan A = \frac{1}{\cot A}, \quad \cot A = \frac{1}{\tan A}$$

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🔸 Pythagorean Identities

Using the Pythagorean Theorem: $$BC^2 + AB^2 = AC^2$$

$$\sin^2 A + \cos^2 A = \frac{BC^2}{AC^2} + \frac{AB^2}{AC^2} = \frac{BC^2 + AB^2}{AC^2} = \frac{AC^2}{AC^2} = 1$$

$$\sec^2 A – \tan^2 A = \frac{AC^2}{AB^2} – \frac{BC^2}{AB^2} = \frac{AC^2 – BC^2}{AB^2} = \frac{AB^2}{AB^2} = 1$$

$$\mathrm{cosec}^2 A – \cot^2 A = \frac{AC^2}{BC^2} – \frac{AB^2}{BC^2} = \frac{AC^2 – AB^2}{BC^2} = \frac{BC^2}{BC^2} = 1$$

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🔸 Alternate Forms of Identities

From the above identities, we get:

$$\sin^2 A = 1 – \cos^2 A \quad ; \quad \cos^2 A = 1 – \sin^2 A$$

$$\sec^2 A = 1 + \tan^2 A \quad ; \quad \tan^2 A = \sec^2 A – 1$$

$$\mathrm{cosec}^2 A = 1 + \cot^2 A \quad ; \quad \cot^2 A = \mathrm{cosec}^2 A – 1$$

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✅ Quotient Trigonometric Identities:

Using the basic definitions of sine and cosine:

$$tan A = \dfrac{\sin A}{\cos A}$$

$$\cot A = \dfrac{\cos A}{\sin A}$$

These are derived from the ratios:

$$\sin A = \dfrac{\text{side opposite to angle } A}{\text{hypotenuse}}$$

$$\cos A = \dfrac{\text{side adjacent to angle } A}{\text{hypotenuse}}$$

So when we divide:

$$\tan A = \dfrac{\sin A}{\cos A} = \dfrac{\dfrac{\text{opposite}}{\text{hypotenuse}}}{\dfrac{\text{adjacent}}{\text{hypotenuse}}} = \dfrac{\text{opposite}}{\text{adjacent}}$$

Similarly: $$\cot A = \dfrac{\cos A}{\sin A} = \dfrac{\dfrac{\text{adjacent}}{\text{hypotenuse}}}{\dfrac{\text{opposite}}{\text{hypotenuse}}} = \dfrac{\text{adjacent}}{\text{opposite}}$$

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Formulas of Basic Trigonometric Ratios

The formulas of these basic trigonometric ratios are listed below:

• Sin θ = Perpendicular / Hypotenuse
• Cos θ = Base / Hypotenuse
• Tan θ = Sin θ / Cos θ = Perpendicular / Base
• Cosec θ = 1 / Sin θ = Hypotenuse / Perpendicular
• Sec θ = 1 / Cos θ = Hypotenuse / Base
• Cot θ = 1 / Tan θ = Base / Perpendicular

List of All Trigonometric Identities

The three basic trigonometric identities are listed below:

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

Some Other Trigonometric Identities

Reciprocal Identities

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ

Quotient Identities

• tan θ = sin θ/cos θ
• cot θ = cos θ/sin θ

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Trigonmetric Identities Table Summary

Identities Name	Identities
Pythagorean Identities	sin2θ + cos2θ = 1
	1 + tan2θ = sec2θ
	1 + cot2θ = cosec2θ
Reciprocal Identities	cosec θ = 1/sin θ
	sec θ = 1/cos θ
	cot θ = 1/tan θ
Quotient Identities	tan θ = sin θ/cos θ
	cot θ = cos θ/sin θ

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Solved Examples on Trigonometric Identities for Class 10

Example 1: Prove that: (1 + cot²A) sin²A = 1

Solution:

LHS = (1 + cot²A) sin²A

Using trigonometric identity

1 + cot²θ = cosec²θ

⇒ LHS = cosec²A sin²A

⇒ LHS = (1 / sin²A) sin²A [cosec A = 1 / sin A]

⇒ LHS = 1

Thus, LHS = RHS

Hence Proved

Example 2: Prove that: tan²B cos²B = 1 – cos²B

Solution:

LHS = tan²B cos²B

Using trigonometric identity

⇒ 1 + tan²θ = sec²θ

⇒ tan²θ = sec²θ – 1

⇒ LHS = (sec²B – 1) cos²B

⇒ LHS = sec²Bcos²B – cos²B

⇒ LHS = [(1/ cos²B)cos²B] – cos²B

⇒ LHS = 1 – cos²B

Thus, LHS = RHS

Hence Proved

Example 3: Prove that: tan θ + (1/tan θ) = sec θ cosec θ

Solution:

LHS = tan θ + (1/tan θ)

⇒ LHS = (tan²θ + 1)/tan θ

Using trigonometric identity

1 + tan²θ = sec²θ

⇒ LHS = sec²θ/tanθ

⇒ LHS = (1 / cos²θ) × (cos θ / sinθ )

⇒ LHS = (1 / cos θ) × (1 / sin θ)

⇒ LHS = sec θ cosec θ [sec θ = 1/cos θ and cosec θ = 1/sin θ]

Thus, LHS = RHS

Hence Proved

Example 4: Prove that: √[(1 – cos θ)/(1 + cos θ)] = cosec θ – cot θ

Solution:

LHS = √[(1 – cos θ)/(1 + cos θ)]

⇒ LHS = √[{(1 – cos θ)/(1 + cos θ)} × {(1 – cos θ) / (1 – cos θ)}]

⇒ LHS = √[(1 – cos θ)²/(1 – cos²θ)]

Using trigonometric identity

sin²θ + cos²θ = 1

⇒ LHS = √[(1 – cosθ)² / sin²θ]

⇒ LHS = (1 – cos θ) / sin θ

⇒ LHS = (1 / sin θ) – (cos θ / sin θ)

⇒ LHS = cosec θ – cot θ [cotθ = cosθ / sinθ and cosecθ = 1/sinθ]

Thus, LHS = RHS

Hence Proved

Example 5: Prove that: sin θ / (1 – cos θ) = cosec θ + cot θ

Solution:

LHS = sin θ/(1 – cos θ)

⇒ LHS = [sin θ/(1 – cos θ)] × [(1 + cos θ)/(1 + cos θ)}]

⇒ LHS = [sin θ (1 + cos θ)] / (1 – cos²θ)

Using trigonometric identity

sin²θ + cos²θ = 1

⇒ LHS = [sin θ (1 + cos θ)] / sin²θ

⇒ LHS = (1 + cos θ)/sin θ

⇒ LHS = (1 / sin θ) + (cos θ/sinθ)

⇒ LHS = cosec θ + cot θ [cot θ = cos θ/sin θ and cosec θ = 1/sin θ]

Thus, LHS = RHS

Hence Proved

Example 6: Prove that: [(1 + cot²θ)tan θ]/sec²θ = cot θ

Solution:

LHS = [(1 + cot²θ)tan θ]/sec²θ

Using trigonometric identity

1 + cot²θ = cosec²θ

⇒ LHS = [cosec²θ tan θ]/sec²θ

⇒ LHS = (1/sin²θ) × (sin θ/cos θ) × (cos²θ/1)

⇒ LHS = (cos θ/ sin θ)

⇒ LHS = cot θ [cot θ = cos θ/sinθ]

Thus, LHS = RHS

Hence Proved

Practice Questions on Trigonometric Identities Class 10

Q1: Prove that: cosecθ√[(1 – cos²θ) = 1

Q2: Prove that: (sec²x – 1)(cosec²x – 1) = 1

Q3: Prove that: sin²C + [1 / (1 + tan²C)] = 1

Q4: Prove that: sec A (1 – sin A) (sec A + tan A) = 1

Q5: Prove that: (1 – sin θ) / (1 + sin θ) = (sec θ – tanθ)²

Q6: Prove that: (cosec θ + sin θ)(cosec θ – sin θ) = cot²θ + cos²θ

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Here is a clear and organized explanation of Frequently Asked Questions (FAQs) on Trigonometry — perfect for academic use or adding to your educational platform like ANAND CLASSES:

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Frequently Asked Questions (FAQs) on Trigonometry

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Q1. What do you mean by Trigonometry?

Answer:
Trigonometry is a branch of mathematics that studies the relationship between the angles and sides of right-angled triangles. It is mainly used to find unknown values using known ratios of sides and angles.
It involves six trigonometric functions that define these relationships.
➡️ Learn more at https://anandclasses.co.in/

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Q2. What are the six basic Trigonometric Functions?

Answer:
The six basic trigonometric functions are:

1. Sine (sin)
2. Cosine (cos)
3. Tangent (tan)
4. Cosecant (cosec)
5. Secant (sec)
6. Cotangent (cot)

These functions are based on the sides of a right triangle and are used to compute angles or lengths in geometric problems.

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Q3. What is the formula for six trigonometric functions?

Answer:
In a right-angled triangle with angle A:

$$\sin A = \dfrac{\text{Opposite side}}{\text{Hypotenuse}}$$

$$\cos A = \dfrac{\text{Adjacent side}}{\text{Hypotenuse}}$$

$$\tan A = \dfrac{\text{Opposite side}}{\text{Adjacent side}}$$

$$\cot A = \dfrac{\text{Adjacent side}}{\text{Opposite side}}$$

$$\sec A = \dfrac{\text{Hypotenuse}}{\text{Adjacent side}}$$

$$\mathrm{cosec}\;A = \dfrac{\text{Hypotenuse}}{\text{Opposite side}}$$

These formulas are fundamental in solving trigonometric problems.

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Q4. What is the primary function of trigonometry?

Answer:
The three primary trigonometric functions are:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)

All other trigonometric functions can be derived from these three. They form the foundation of trigonometry.

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Q5. Who is the founder of trigonometry?

Answer:
Hipparchus, a Greek astronomer, geographer, and mathematician, is credited with discovering the early concepts of trigonometry around 150 B.C. He compiled the first known trigonometric table.

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Q6. What are the Applications of Trigonometry in Real Life?

Answer:
Trigonometry has numerous real-world applications, including:

• Measuring Heights and Distances (e.g., mountains, towers)
• Aviation and Aeronautics
• Navigation (Air and Sea)
• Criminology (e.g., determining trajectories)
• Engineering and Architecture
• Marine Biology

These fields use trigonometric calculations to solve practical problems involving angles and measurements.

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✅ MCQs on Trigonometry with Answers & Explanations

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Q1. Which of the following is NOT a trigonometric function?

A. Sine
B. Cosine
C. Logarithm
D. Tangent

Answer: ✅ C. Logarithm
Explanation: Logarithm is a mathematical function but not a trigonometric one. Trigonometric functions include sine, cosine, tangent, etc.

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Q2. If $\sin A = \dfrac{3}{5}$, and A is an acute angle, what is the value of cos A?

A. $\dfrac{4}{5}$
B. $\dfrac{5}{3}$
C. $\dfrac{3}{4}$
D. $\dfrac{4}{3}$

Answer: ✅ A. $\dfrac{4}{5}$
Explanation:
Use identity:
$$\sin^2 A + \cos^2 A = 1 $$

$$\left( \dfrac{3}{5} \right)^2 + \cos^2 A = 1 \Rightarrow \cos^2 A = 1 – \dfrac{9}{25} = \dfrac{16}{25} \Rightarrow \cos A = \dfrac{4}{5}$$

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Q3. What is the value of $\tan 45^\circ$?

A. 0
B. 1
C. $\sqrt{3}$
D. Not defined

Answer: ✅ B. 1
Explanation:
At $45^\circ$, the opposite and adjacent sides of a right triangle are equal. Hence,
$\tan 45^\circ = \dfrac{\text{Opposite}}{\text{Adjacent}} = 1$

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Q4. If $\cot A = \dfrac{1}{\sqrt{3}}$, then what is the value of A?

A. $30^\circ$
B. $45^\circ$
C. $60^\circ$
D. $90^\circ$

Answer: ✅ C. $60^\circ$
Explanation:
We know $\cot 60^\circ = \dfrac{1}{\sqrt{3}}$, so $A = 60^\circ$

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Q5. Which of the following is the correct value of $\mathrm{cosec}\;30^\circ$?

A. 1
B. 2
C. 2\sqrt{2}
D. 232\sqrt{3}

Answer: ✅ B. 2
Explanation: $\mathrm{cosec}\;30^\circ = \dfrac{1}{\sin 30^\circ} = \dfrac{1}{1/2} = 2$

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Q6. Which identity is always true for all values of angle θ\theta?

A. $\sin^2 \theta + \cos^2 \theta = 1$
B. $\tan \theta + \cot \theta = 1$
C. $\sec \theta + \mathrm{cosec} \theta = 1$
D. $\tan^2 \theta = \cot^2 \theta$

Answer: ✅ A. $\sin^2 \theta + \cos^2 \theta = 1$
Explanation: This is the fundamental Pythagorean identity in trigonometry.

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Q7. If $\tan A = 1$, then the value of A is—

A. $30^\circ$
B. $45^\circ$
C. $60^\circ$
D. $90^\circ$

Answer: ✅ B. $45^\circ$
Explanation: $\tan 45^\circ$ = 1, hence A = $45^\circ$

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🧠 “Do You Know?” Facts about Trigonometry

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🔹 The word “Trigonometry” comes from Greek.
It is derived from the Greek words “trigonon” (triangle) and “metron” (measure), meaning “measuring triangles.”

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🔹 Trigonometry was originally developed for astronomy.
Ancient astronomers used trigonometric concepts to study the motion of celestial bodies and map the night sky.

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🔹 Right-angled triangles are the base of all trigonometric studies.
Even complex applications in calculus and physics often trace back to the simple ratios found in right triangles.

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🔹 Sine and Cosine functions form the basis of sound waves and signals.
These functions are used in music, electronics, and communication systems like mobile phones and radios.

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🔹 The first trigonometric table was created over 2000 years ago.
Hipparchus created a chord table, which is considered the earliest form of a trigonometric table.

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🔹 Your GPS uses trigonometry!
Global Positioning Systems (GPS) rely on the principles of trigonometry to calculate positions and distances based on satellite signals.

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🔹 Trigonometry helps build roller coasters!
Engineers use it to design the slopes, curves, and loops for thrilling rides with accurate safety measures.

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🔹 Bees unknowingly use trigonometry!
The hexagonal shape of honeycombs, which involves precise angles, is a result of natural optimization — a concept deeply connected to geometry and trigonometry.

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🔹 Trigonometry is essential in video game design.
From the way characters move to how shadows are cast — trigonometric calculations power a lot of the visual realism in gaming.

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