Introduction to Trigonometry — Class 10 Mathematics

"Triangles + Ratios = Trigonometry. The mathematics that measures the sky."

1. About the Chapter

Trigonometry = 'measurement of triangles' (Greek: 'trigonon' + 'metron'). Used in:

• Astronomy (measuring stars)
• Surveying (measuring heights)
• Engineering (forces, designs)
• Physics (waves, oscillations)

This chapter introduces the six trigonometric ratios and basic identities.

Indian Heritage

• Aryabhata (5th century CE): defined sine ('jya'); coined trigonometric ratios
• Brahmagupta, Bhaskara II: refined and extended trigonometry
• Sanskrit 'jya' became Arabic 'jiba' became Latin 'sinus' → English 'sine'

2. Right-Angled Triangle Review

For △ABC with right angle at B:

         A
         /|
        / |
       /  | ← perpendicular (P)
      /θ_|
     B    C
     base (B)
   hypotenuse from B to A

Wait, let me redraw. In a right triangle with right angle at C:

• Side opposite to angle A = a (perpendicular for angle A)
• Side opposite to angle B = b
• Side opposite to right angle (C) = c (hypotenuse)

For ANGLE θ (one of the acute angles):

• Opposite side: side opposite to θ
• Adjacent side: side next to θ (not hypotenuse)
• Hypotenuse: longest side, opposite to 90° angle

3. The Six Trigonometric Ratios

For an acute angle θ in a right triangle:

Sine (sin θ)

sin θ = Opposite / Hypotenuse = P / H

Cosine (cos θ)

cos θ = Adjacent / Hypotenuse = B / H

Tangent (tan θ)

tan θ = Opposite / Adjacent = P / B

Cosecant (cosec θ) — reciprocal of sin

cosec θ = Hypotenuse / Opposite = H / P = 1 / sin θ

Secant (sec θ) — reciprocal of cos

sec θ = Hypotenuse / Adjacent = H / B = 1 / cos θ

Cotangent (cot θ) — reciprocal of tan

cot θ = Adjacent / Opposite = B / P = 1 / tan θ

Memory Aid (SOH CAH TOA)

• SOH: Sin = Opposite/Hypotenuse
• CAH: Cos = Adjacent/Hypotenuse
• TOA: Tan = Opposite/Adjacent

4. Reciprocal Relations

These give 3 more ratios:

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

Also:

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

5. Trigonometric Ratios of Specific Angles

TABLE (MEMORISE!)

Memory Tricks

• sin 0°, 30°, 45°, 60°, 90° = √0/2, √1/2, √2/2, √3/2, √4/2
• That is: 0, 1/2, √2/2, √3/2, 1

cos is sin in REVERSE order:

• cos 0°, 30°, 45°, 60°, 90° = 1, √3/2, √2/2, 1/2, 0

tan = sin / cos (gets ∞ where cos = 0)

6. Pythagoras and Trigonometry

From Pythagoras: P² + B² = H²

Divide by H²: (P/H)² + (B/H)² = 1

This gives: sin²θ + cos²θ = 1 — fundamental identity!

7. Trigonometric Identities (MEMORISE)

Three Key Identities

Identity 1: sin²θ + cos²θ = 1

Identity 2: 1 + tan²θ = sec²θ

Identity 3: 1 + cot²θ = cosec²θ

Derivations

Identity 2: Divide Identity 1 by cos²θ:

• sin²θ/cos²θ + 1 = 1/cos²θ
• tan²θ + 1 = sec²θ → 1 + tan²θ = sec²θ ✓

Identity 3: Divide Identity 1 by sin²θ:

• 1 + cos²θ/sin²θ = 1/sin²θ
• 1 + cot²θ = cosec²θ ✓

Useful Forms

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

8. Trigonometric Ratios of Complementary Angles

If A + B = 90° (complementary):

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

Examples

• sin 30° = cos 60° = 1/2 ✓
• tan 45° = cot 45° = 1 ✓
• sec 60° = cosec 30° = 2 ✓

9. Worked Examples

Example 1: Find Ratios

In right △ABC, ∠B = 90°, AB = 3, BC = 4. Find sin C, cos C, tan C.

• AC (hypotenuse) = √(9+16) = 5
• For angle C: opposite = AB = 3, adjacent = BC = 4, hypotenuse = 5
• sin C = 3/5, cos C = 4/5, tan C = 3/4

Example 2: Use Identity

Find sin A if cos A = 5/13.

• sin²A + cos²A = 1
• sin²A = 1 − 25/169 = 144/169
• sin A = 12/13

Example 3: Specific Values

Evaluate: sin 30° × cos 60° + sin 60° × cos 30°

• = (1/2)(1/2) + (√3/2)(√3/2)
• = 1/4 + 3/4 = 1

(This is actually sin(30° + 60°) = sin 90° = 1, using formula sin(A+B), but for Class 10 we calculate directly.)

Example 4: Prove Identity

Prove: (1 − sin²θ) × sec²θ = 1

• LHS = cos²θ × (1/cos²θ) = 1
• = RHS ✓

Example 5: Complementary Angles

sin 60° = cos ?

• 60° = 90° − 30°
• sin 60° = cos 30°
• So the answer is 30°

10. Common Mistakes

1. Confusing opposite and adjacent

   • 'Opposite' is OPPOSITE to the angle θ; 'Adjacent' is the OTHER side (not hypotenuse).

2. Wrong specific angle values

   • MEMORISE the table. Don't guess.

3. Forgetting reciprocal relations

   • cosec ≠ cos. cosec = 1/sin.

4. Identity confusion

   • sin²θ + cos²θ = 1 (NOT sin θ + cos θ = 1).

5. tan at 90°

   • tan 90° is UNDEFINED (∞), not 0.

11. Indian Heritage of Trigonometry

Aryabhata (476-550 CE)

• Defined SINE (jya), COSINE (kojya), VERSINE
• Tables of sines for various angles
• Used in astronomy

Brahmagupta (598-668 CE)

• Extended Aryabhata's work
• Brahmagupta-Fibonacci identity

Bhaskara II (1114-1185 CE)

• Refined trigonometric calculations
• Indian astronomy depended heavily on this

Madhava (1340-1425 CE)

• INFINITE SERIES for sin, cos, arctan
• 200 years before Newton!
• Astonishing achievement

Word Origin

Sanskrit 'jya' → Arabic 'jiba' → Latin 'sinus' → English 'sine'

A direct line from Indian mathematics to global use.

12. Conclusion

Trigonometry is one of the MOST USED branches of mathematics:

• Astronomy: distances to stars, planet orbits
• Engineering: forces, structures, machines
• Physics: waves, oscillations, optics
• Computer graphics: rotations, animations

Master:

• 6 trigonometric ratios (SOH CAH TOA)
• Specific angle values (0°, 30°, 45°, 60°, 90°)
• 3 identities (sin² + cos² = 1, etc.)
• Complementary angles relations

Chapter 9 will apply these to REAL-WORLD problems (heights and distances).

Trigonometry: the mathematics of triangles, taught by Indians to the world.