Polytechnic 1st semester mathematics trigonometrical ratios of allied angle

Chapter 2: Trigonometric Ratios of Allied Angles

Subject: Mathematics
Course: Polytechnic 1st Semester
Unit: Trigonometry

---

🔹 Introduction

In previous chapter, we studied trigonometric ratios of acute angles (less than 90°). In this chapter, we extend trigonometric ratios to all angles such as 0°, 30°, 45°, 60°, 90°, 180°, 270° and 360°.

This concept is very important for:

• Board Exams
• Engineering Mathematics
• Higher studies in calculus and physics

---

🔹 Trigonometric Ratios

The six basic trigonometric ratios are:

• sin θ (Sine)
• cos θ (Cosine)
• tan θ (Tangent)
• cosec θ (Cosecant)
• sec θ (Secant)
• cot θ (Cotangent)

📌 Definitions (Right Angled Triangle)

• sin θ = Perpendicular / Hypotenuse
• cos θ = Base / Hypotenuse
• tan θ = Perpendicular / Base
• cosec θ = Hypotenuse / Perpendicular
• sec θ = Hypotenuse / Base
• cot θ = Base / Perpendicular

---

🔹 Trigonometric Ratios of Standard Angles

📊 Table of Trigonometric Ratios

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	∞
cosec θ	∞	2	√2	2/√3	1
sec θ	1	2/√3	√2	2	∞
cot θ	∞	√3	1	1/√3	0

---

🎥 Video Lecture – Trigonometric Ratios Table

---

🔹 Trigonometric Ratios of 180° and 360°

Using unit circle concept:

• sin 180° = 0
• cos 180° = −1
• tan 180° = 0
• sin 360° = 0
• cos 360° = 1
• tan 360° = 0

---

🎥 Video Lecture – Trigonometric Ratios of 180° & 360°

---

🔹 Trigonometric Ratios of 270°

• sin 270° = −1
• cos 270° = 0
• tan 270° = ∞

---

🎥 Video Lecture – Trigonometric Ratios of 270°

---

🔹 Important Trigonometric Identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ
• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ

---

🎥 Video Lecture – Trigonometric Identities

---

🔹 Solved Examples

Example 1:

Find the value of sin 60° + cos 30°

Solution:
sin 60° = √3/2
cos 30° = √3/2
Answer = √3

Example 2:

Evaluate: tan 45° × cos 60°

tan 45° = 1
cos 60° = 1/2
Answer = 1/2

---

🎥 Video Lecture – Solved Examples

---

🔹 Exam Tips & Memory Tricks

• Remember sin values increase from 0° to 90°
• cos values decrease from 0° to 90°
• tan 90° is not defined
• Use identities to simplify expressions

---

✅ Chapter Summary

In this chapter, we studied:

• Trigonometric ratios of all angles
• Standard values
• Important identities
• Numerical problems

✔ This chapter is very important for exams.