TRIGONOMETRY COMPLETE GUIDE
Detailed Notes, Diagrams, Formulas, Examples & Exercise
📚 Contents
1. Introduction to Trigonometry
Trigonometry is a branch of mathematics that studies the relationship between the sides and angles of a triangle. It is widely used in engineering, astronomy, navigation, construction and physics.
Trigonometry = Triangle + Measurement
It mainly deals with right-angled triangles.
2. Right Angled Triangle
A triangle having one angle equal to 90° is called a right triangle.
Parts of Right Triangle
- Hypotenuse: Longest side opposite to 90°
- Base: Adjacent side
- Perpendicular: Vertical side
3. Trigonometric Ratios
The six trigonometric ratios are:
| Ratio | Formula |
|---|---|
| sin θ | Perpendicular / Hypotenuse |
| cos θ | Base / Hypotenuse |
| tan θ | Perpendicular / Base |
| cosec θ | Hypotenuse / Perpendicular |
| sec θ | Hypotenuse / Base |
| cot θ | Base / Perpendicular |
sin θ = P/H
cos θ = B/H
tan θ = P/B
cos θ = B/H
tan θ = P/B
Important Relation
tan θ = sin θ / cos θ
Example 1
In a triangle:
Perpendicular = 3 cm
Base = 4 cm
Hypotenuse = 5 cm
Find sin θ, cos θ and tan θ.
Perpendicular = 3 cm
Base = 4 cm
Hypotenuse = 5 cm
Find sin θ, cos θ and tan θ.
sin θ = 3/5
cos θ = 4/5
tan θ = 3/4
cos θ = 4/5
tan θ = 3/4
4. Standard Trigonometric Values
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | ∞ |
Example 2
Find the value of:
sin 30° + cos 60°
sin 30° + cos 60°
sin 30° = 1/2
cos 60° = 1/2
Answer = 1/2 + 1/2 = 1
cos 60° = 1/2
Answer = 1/2 + 1/2 = 1
5. Complete Trigonometric Identities
Trigonometric identities are equations that remain true for every value of angle θ. These identities are used to simplify expressions, solve equations, and prove formulas.
1. Reciprocal Identities
sin θ = 1/cosec θ
cos θ = 1/sec θ
tan θ = 1/cot θ
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
Example:
If cosec θ = 4, find sin θ.
If cosec θ = 4, find sin θ.
sin θ = 1/cosec θ
= 1/4
= 1/4
2. Quotient Identities
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
Example:
If sin θ = 3/5 and cos θ = 4/5, find tan θ.
If sin θ = 3/5 and cos θ = 4/5, find tan θ.
tan θ = sin θ / cos θ
= (3/5)/(4/5)
= 3/4
= (3/5)/(4/5)
= 3/4
3. Pythagorean Identities
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ
Example:
If tan θ = 3, find sec θ.
If tan θ = 3, find sec θ.
1 + tan² θ = sec² θ
1 + 3² = sec² θ
1 + 9 = sec² θ
sec² θ = 10
sec θ = √10
1 + 3² = sec² θ
1 + 9 = sec² θ
sec² θ = 10
sec θ = √10
4. Complementary Angle Identities
sin(90° − θ) = cos θ
cos(90° − θ) = sin θ
tan(90° − θ) = cot θ
cot(90° − θ) = tan θ
sec(90° − θ) = cosec θ
cosec(90° − θ) = sec θ
Example:
Find:
sin 60°
Find:
sin 60°
sin(90° − θ) = cos θ
sin 60°
= cos 30°
= √3/2
sin 60°
= cos 30°
= √3/2
5. Negative Angle Identities
sin(−θ) = −sin θ
cos(−θ) = cos θ
tan(−θ) = −tan θ
cot(−θ) = −cot θ
sec(−θ) = sec θ
cosec(−θ) = −cosec θ
Example:
Find:
cos(−45°)
Find:
cos(−45°)
cos(−θ) = cos θ
cos(−45°)
= cos45°
= 1/√2
cos(−45°)
= cos45°
= 1/√2
6. Double Angle Identities
sin 2θ = 2 sin θ cos θ
cos 2θ = cos² θ − sin² θ
cos 2θ = 2cos² θ − 1
cos 2θ = 1 − 2sin² θ
tan 2θ = 2tan θ / (1 − tan² θ)
Example:
If sin θ = 1/2 and cos θ = √3/2, find sin 2θ.
If sin θ = 1/2 and cos θ = √3/2, find sin 2θ.
sin 2θ = 2 sin θ cos θ
= 2 × (1/2) × (√3/2)
= √3/2
= 2 × (1/2) × (√3/2)
= √3/2
7. Half Angle Identities
sin θ/2 = √[(1 − cos θ)/2]
cos θ/2 = √[(1 + cos θ)/2]
tan θ/2 = sin θ / (1 + cos θ)
tan θ/2 = (1 − cos θ)/sin θ
Example:
Find cos30° using half-angle identity.
Find cos30° using half-angle identity.
cos30° = cos(60°/2)
= √[(1 + cos60°)/2]
= √[(1 + 1/2)/2]
= √(3/4)
= √3/2
= √[(1 + cos60°)/2]
= √[(1 + 1/2)/2]
= √(3/4)
= √3/2
8. Sum and Difference Identities
sin(A + B) = sinA cosB + cosA sinB
sin(A − B) = sinA cosB − cosA sinB
cos(A + B) = cosA cosB − sinA sinB
cos(A − B) = cosA cosB + sinA sinB
tan(A + B) =
(tanA + tanB)/(1 − tanA tanB)
tan(A − B) =
(tanA − tanB)/(1 + tanA tanB)
Example:
Find:
sin(45° + 45°)
Find:
sin(45° + 45°)
sin(A + B)
= sin45° cos45° + cos45° sin45°
= (1/√2 × 1/√2) + (1/√2 × 1/√2)
= 1/2 + 1/2
= 1
= sin45° cos45° + cos45° sin45°
= (1/√2 × 1/√2) + (1/√2 × 1/√2)
= 1/2 + 1/2
= 1
9. Product-to-Sum Identities
sinA sinB =
½[cos(A − B) − cos(A + B)]
cosA cosB =
½[cos(A − B) + cos(A + B)]
sinA cosB =
½[sin(A + B) + sin(A − B)]
Example:
Simplify:
2 sinA cosA
Simplify:
2 sinA cosA
Using:
sin2A = 2 sinA cosA
Therefore:
2 sinA cosA = sin2A
sin2A = 2 sinA cosA
Therefore:
2 sinA cosA = sin2A
10. Important Standard Results
sin0° = 0
cos0° = 1
tan45° = 1
sin90° = 1
cos90° = 0
tan90° = ∞
11. Uses of Trigonometric Identities
- Simplifying trigonometric expressions
- Solving equations
- Finding unknown sides and angles
- Heights and distances problems
- Engineering calculations
- Astronomy and navigation
- Wave and sound calculations in physics
- Architecture and construction
6. Heights and Distances
Trigonometry helps to find heights and distances using angles.
Angle of Elevation
When we look upward at an object, the angle formed is called angle of elevation.
Example 4
A tower is 20 m high.
The angle of elevation from a point is 45°.
Find the distance from the point to the tower.
tan 45° = Height / Distance
1 = 20 / Distance
Distance = 20 m
1 = 20 / Distance
Distance = 20 m
7. Practice Exercises
Exercise 1
Find:1. sin 60°
2. cos 30°
3. tan 45°
1. sin 60° = √3/2
2. cos 30° = √3/2
3. tan 45° = 1
2. cos 30° = √3/2
3. tan 45° = 1
Exercise 2
If perpendicular = 5 cm and hypotenuse = 13 cm, find sin θ.
sin θ = P/H
= 5/13
= 5/13
Exercise 3
Prove:sin² θ + cos² θ = 1
Using right triangle:
sin θ = P/H
cos θ = B/H
sin² θ + cos² θ
= P²/H² + B²/H²
= (P² + B²)/H²
Using Pythagoras:
P² + B² = H²
Therefore:
= H²/H² = 1
sin θ = P/H
cos θ = B/H
sin² θ + cos² θ
= P²/H² + B²/H²
= (P² + B²)/H²
Using Pythagoras:
P² + B² = H²
Therefore:
= H²/H² = 1
🎯 Quick Formula Revision
sin θ = P/H
cos θ = B/H
tan θ = P/B
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ