Trigonometry

Trigonometry

Trigonometry is a branch of mathematics that studies the relationship between the angles and sides of a triangle—especially right-angled triangles. It is widely used in physics, engineering, architecture, navigation, astronomy and everyday problem-solving.
The core of trigonometry lies in the three most important ratios:

• Sine (sin) = Perpendicular / Hypotenuse
• Cosine (cos) = Base / Hypotenuse
• Tangent (tan) = Perpendicular / Base

Understanding these functions helps us measure heights, distances, angles, and the behavior of waves, light, sound, and motion. Trigonometry forms a foundation for advanced mathematics and scientific studies.

Basic Trigonometric Ratios

1. sin 0° = 0
2. sin 30° = 1/2
3. sin 45° = √2/2
4. sin 60° = √3/2
5. sin 90° = 1
6. cos 0° = 1
7. cos 30° = √3/2
8. cos 45° = √2/2
9. cos 60° = 1/2
10. cos 90° = 0
11. tan 0° = 0
12. tan 30° = 1/√3
13. tan 45° = 1
14. tan 60° = √3
15. tan 90° = Undefined

Reciprocal Ratios

1. cosec 30° = 2
2. sec 60° = 2
3. cot 45° = 1
4. cosec 90° = 1
5. cot 0° = Undefined

Pythagorean Identities

1. If sin θ = 3/5, find cos θ (acute angle).
   cos θ = 4/5.
2. If cos θ = 12/13, find sin θ.
   sin θ = 5/13.
3. If tan θ = 4/3, find sec θ.
   sec θ = 5/3.

Height and Distance Problems

1. A ladder 10 m long makes a 60° angle with the ground.
   Height = 10 × sin 60° = 10 × √3/2 = 5√3 m.
2. A tree casts a 20 m shadow when the sun’s angle of elevation is 30°.
   Height = 20 × tan 30° = 20 × 1/√3 = 20/√3 m.
3. A building is 40 m tall. Angle of elevation from a point is 45°.
   Distance = 40 m.
4. A person looks at the top of a tower at 60°, distance 30 m.
   Height = 30 × tan 60° = 30√3 m.
5. Height of a tower is 50 m. Angle of depression is 30°.
   Distance = 50 / tan 30° = 50√3 m.

Finding Unknown Sides

1. In a right triangle, hypotenuse = 13, angle = 30°.
   Opposite side = 13 × sin 30° = 6.5.
2. Hypotenuse = 10, base = 8.
   Perpendicular = √(10² − 8²) = 6.
3. Base = 7, angle = 45°.
   Opposite = 7 × tan 45° = 7.
4. If sin θ = 1/2, angle θ = 30°.
5. If cos θ = √3/2, θ = 30°.

Trigonometric Identities

1. sin² 30° + cos² 30° = 1
2. tan 45° × tan 45° = 1
3. cos 60° × sec 60° = 1
4. sin 90° × cosec 90° = 1
5. tan 30° × cot 30° = 1

Angle Simplifications

1. sin (90° – 30°) = cos 30° = √3/2
2. cos (90° – 45°) = sin 45° = √2/2
3. tan (90° – 60°) = cot 60° = 1/√3

Trigonometric Values Using Triangles

1. Triangle with sides 5, 12, 13. Find sin θ opposite to side 5.
   sin θ = 5/13
2. Same triangle, find cos θ for adjacent side 12.
   cos θ = 12/13
3. Same triangle, find tan θ.
   tan θ = 5/12

Advanced Examples

1. If sin A = 4/5, find tan A.
   Perpendicular = 4, Hyp = 5 → Base = 3
   tan A = 4/3
2. If cos A = 3/5, find tan A.
   Perpendicular = 4, Base = 3
   tan A = 4/3
3. If cot A = 5/12, find sin A.
   Base = 12, Perpendicular = 5 → Hypotenuse = 13
   sin A = 5/13
4. If sec A = 5/4, find cos A.
   cos A = 4/5
5. If tan A = √3, angle A = 60°.
6. If sin θ = cos θ,
   θ = 45°.

Trigonometry is an essential mathematical tool that connects angles and distances. By mastering trigonometric ratios, identities, and height-distance applications, students can confidently solve real-life problems related to engineering, navigation, physics, architecture, and surveying. A strong understanding of trigonometry builds analytical thinking, accuracy, and a solid foundation for advanced mathematical concepts.

FAQs

1. What is trigonometry mainly used for?

It is used to measure heights, distances, angles, and analyze waves, motion, light and engineering structures.

2. What are the three basic trigonometric ratios?

Sine, Cosine, and Tangent.

3. Why is trigonometry important for students?

It improves logical thinking and helps in science, engineering, and real-world calculations.

4. What is the easiest way to learn trigonometry?

Start with memorizing standard angle values and understanding right-triangle ratios.

5. Where is trigonometry used in real life?

In navigation, architecture, astronomy, physics, GPS technology, and construction.

6. What is the meaning of 0° and 90° in trigonometry?

They represent special angles with standard trigonometric values.

7. Can trigonometry be used without triangles?

Yes, it is used in waves, circular motion, oscillations, and periodic patterns.

8. What is the most important identity?

sin²θ + cos²θ = 1

9. Why is tan 90° undefined?

Because cos 90° = 0, and division by zero is impossible.

10. Should I memorize trigonometric tables?

Yes, knowing standard angles (0°, 30°, 45°, 60°, 90°) is essential.

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