Chapter – 8

Introduction to Trigonometry

In this post we have given the detailed notes of class 10 Maths Chapter 8 (Introduction to Trigonometry) in English. These notes are useful for the students who are going to appear in class 10 board exams.

Chapter 8: Introduction to Trigonometry

What is Trigonometry?

The word ‘Trigonometry’ is derived from three Greek words: ‘tri’ (three), ‘gon’ (sides) and ‘metron’ (measure) — meaning “measurement of the sides of a triangle.” Trigonometry is the branch of mathematics that deals with the relationship between the sides and angles of a right-angled triangle. Ancient civilisations in Egypt and Babylon are known to have worked on early ideas of trigonometry, mainly for use in astronomy.

Important Terms

• Right-angled triangle — a triangle in which one angle measures 90°.
• Acute angle — an angle whose measure is less than 90°.
• In a right triangle, the side opposite the angle under consideration is called the Perpendicular, the side next to the angle (other than the hypotenuse) is called the Base, and the side opposite the right angle is called the Hypotenuse.

Trigonometric Ratios

For an acute angle A in a right triangle, the trigonometric ratios are defined as follows (P = Perpendicular, B = Base, H = Hypotenuse):

Full names of the ratios: sin – sine, cos – cosine, tan – tangent, cosec – cosecant, sec – secant, cot – cotangent.

Reciprocal Relationships

• sin A = 1/cosec A and cosec A = 1/sin A
• cos A = 1/sec A and sec A = 1/cos A
• tan A = 1/cot A and cot A = 1/tan A
• tan A = sin A/cos A
• cot A = cos A/sin A

Note: tan A is not the product of “tan” and “A” — tan has no meaning without A attached to it. The same is true of all other trigonometric ratios. The value of a trigonometric ratio of an angle depends only on the angle, not on the size of the right triangle — the ratios are identical in all similar triangles.

Remember: Since the hypotenuse is always the longest side of a right triangle, the value of sin A and cos A is always less than 1 (or equal to 1 in a limiting case).

Worked Example

Q. If tan A = 4/3, find the other trigonometric ratios of angle A.

Solution: Let us draw a right triangle ABC, right-angled at B. Since tan A = Perpendicular/Base = BC/AB = 4/3, let BC = 4k and AB = 3k, where k is a positive number.

By the Pythagoras Theorem: AC² = AB² + BC² = (4k)² + (3k)² = 16k² + 9k² = 25k², so AC = 5k.

Now, using the definitions of the ratios:

• sin A = BC/AC = 4k/5k = 4/5
• cos A = AB/AC = 3k/5k = 3/5
• tan A = BC/AB = 4k/3k = 4/3
• cosec A = AC/BC = 5k/4k = 5/4
• sec A = AC/AB = 5k/3k = 5/3
• cot A = AB/BC = 3k/4k = 3/4

Trigonometric Ratios of Specific Angles

Ratios of 45°

In △ABC, right-angled at B, if ∠A = 45°, then ∠C = 45° too, so BC = AB. Let BC = AB = a. By the Pythagoras Theorem, AC² = a² + a² = 2a², so AC = a√2.

sin 45° = BC/AC = 1/√2, cos 45° = AB/AC = 1/√2, tan 45° = BC/AB = 1

Ratios of 30° and 60°

Consider an equilateral triangle ABC with each angle = 60°. Draw AD ⊥ BC. Then △ABD ≅ △ACD, so BD = DC and ∠BAD = ∠CAD = 30°.

Let AB = 2a. Then BD = a, and AD² = AB² − BD² = (2a)² − a² = 3a², so AD = a√3.

sin 30° = BD/AB = 1/2, cos 30° = AD/AB = √3/2, tan 30° = BD/AD = 1/√3

sin 60° = AD/AB = √3/2, cos 60° = BD/AB = 1/2, tan 60° = AD/BD = √3

Ratios of 0° and 90°

As ∠A in a right triangle ABC is made smaller and smaller until it approaches 0°, side BC shrinks toward 0 and AC becomes almost equal to AB. This gives sin 0° = 0 and cos 0° = 1.

As ∠A is increased until it approaches 90°, side AB shrinks toward 0 and AC becomes almost equal to BC. This gives sin 90° = 1 and cos 90° = 0.

Table of Trigonometric Ratios of Standard Angles

Observation: As ∠A increases from 0° to 90°, sin A increases from 0 to 1, while cos A decreases from 1 to 0.

Trigonometric Ratios of Complementary Angles

Two angles are complementary if their sum is 90°. In a right triangle ABC, right-angled at B, ∠A + ∠C = 90°, so ∠C = 90° − ∠A.

• sin (90° − A) = cos A
• cos (90° − A) = sin A
• tan (90° − A) = cot A
• cot (90° − A) = tan A
• cosec (90° − A) = sec A
• sec (90° − A) = cosec A

(These hold true for all values of A between 0° and 90°.)

Trigonometric Identities

An equation involving trigonometric ratios of an angle is called a trigonometric identity if it is true for all values of the angle(s) involved. The three fundamental identities are:

• sin² A + cos² A = 1
• 1 + tan² A = sec² A
• 1 + cot² A = cosec² A

These can be rearranged as needed, for example:

• sin² A = 1 − cos² A and cos² A = 1 − sin² A
• tan² A = sec² A − 1
• cot² A = cosec² A − 1

Note: If one trigonometric ratio of an acute angle is known, all the other ratios of that angle can be found — either by using the identities, or by constructing a right triangle and applying the Pythagoras Theorem.

Additional Trigonometric Formulas (For Reference)

The following formulas go beyond the Class 10 NCERT syllabus but are useful for competitive exams and higher classes.

Sign of Trigonometric Ratios by Quadrant

• Quadrant I: all ratios positive
• Quadrant II: only sin and cosec positive
• Quadrant III: only tan and cot positive
• Quadrant IV: only cos and sec positive

Sum and Difference Formulas

• 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) = (tan A + tan B)/(1 − tan A tan B)
• tan (A − B) = (tan A − tan B)/(1 + tan A tan B)

Double Angle Formulas

• sin 2A = 2 sin A cos A
• cos 2A = cos² A − sin² A
• tan 2A = 2 tan A/(1 − tan² A)

Key Points to Remember

• sin A = P/H, cos A = B/H, tan A = P/B, cosec A = H/P, sec A = H/B, cot A = B/P
• tan A = sin A/cos A and cot A = cos A/sin A
• sin² A + cos² A = 1; 1 + tan² A = sec² A; 1 + cot² A = cosec² A
• sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1
• cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, cos 90° = 0
• tan 45° = 1, tan 30° = 1/√3, tan 60° = √3; tan 90° is not defined
• sin (90° − A) = cos A and cos (90° − A) = sin A
• tan (90° − A) = cot A and cot (90° − A) = tan A
• The value of sin A and cos A never exceeds 1; the value of sec A and cosec A is never less than 1.

We hope that class 10 Maths Chapter 8 (Introduction to Trigonometry) notes in English helped you. If you have any query about class 10 Maths Chapter 8 (Introduction to Trigonometry) notes in English or about any other notes of class 10 Maths in English, so you can comment below. We will reach you as soon as possible…